aula de empuxos

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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   1
ESCOLA DE ENGENHARIA DA UFMG
Departamento de Engenharia de Transportes e Geotecnia
Cálculo de Empuxos
1a parte
Mecânica dos Solos II
TURMA C – Prof. Lúcio Villar
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   2
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   3
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   4
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   5
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   6
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   7
Deslizamento
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   8
Coeficientes de empuxo:
1. Coeficiente de empuxo no repouso:
 
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Pela Teoria da Elasticidade:
Onde μ é o coeficiente 
de Poisson do solo
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   9
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   10
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   11
 
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Extensão 
lateral
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   12
 
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Empuxo devido ao peso próprio Empuxo devido à coesão
SOLO COM COESÃO:
aaa kczk 2−=′ γσ
 
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SOLO COM COESÃO:
Trecho com parcela de 
coesão maior que a devido 
ao peso próprio
aaa kczk 2−=′ γσ
Trecho com parcela devido 
ao peso próprio é maior que 
a de coesão
Formação de trincas de tração
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   13
SOLO COM COESÃO:
aaa kczk 2−=′ γσ
γγ a
a
taa k
kc
zkczk
2
2 =∴=
Hcr
t
a
a
cr zk
kc
H 2
4 == γ
 
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Empuxo devido ao peso próprio Empuxo devido à coesão
SOLO COM COESÃO: Ponto de aplicação do empuxo resultante
zkaγ
akc2−
H/3
H/2
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   14
Para solos coesivos
1
1 Peso próprio do solo
22 coesão
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   15
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   16
 
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Para que se produza o deslizamento, o 
empuxo deverá ser maior do que o 
peso do terrapleno, assim, a pressão 
principal maior é a horizontal e a 
menor a vertical.
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   17
 
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Estado inicial de tensão
Trajetória de extensão 
lateral: empuxo ativo
Trajetória de compressão 
lateral: empuxo passivo
 
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α
Não é a tensão normal ao 
plano do elemento de solo
N = Wcosα = γzcosα
σnormal vertical = γzcosα2
α
α
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   18
φ’τ
σC
D αA
B
σ3
σ1
φ’
E
σhcosα
γzcosα
 
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φ’τ
σC
D αA
B
σ3
σ1
φ’
E
σhcosα
γzcosα
φαα
φαα
22
22
coscoscos
coscoscos
−+
−−=ak
αγσ cos)( zkaativoh =
αcos.
(kN/m) cos
2
1 2 αγ aa kHE =
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   19
Resultante de empuxo
Linha de atuação do 
empuxo
α
α
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   20
 
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• considera o atrito entre o solo e muro.
• não exige paramento vertical
• melhor quando se tem muitos esforços envolvidos
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   21
 
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Método das cunhas ou tentativas
1. considere uma superfície planar qualquer que passa pelo pé do muro e define 
uma cunha de solo que deslizará.
2. admita que a superfície de ruptura é plana.
3. pode ser determinado o peso desta cunha, que tem direção conhecida.
4. a direção da resultante de empuxo também é conhecida e faz um ângulo δ com 
a normal ao paramento de montante.
5. o ângulo δ é igual ao ângulo de atrito entre o solo e o muro. Ele pode ser 
estimado, para o caso ativo, como sendo igual a 2/3 do ângulo de atrito interno 
do solo.
6. admite-se que a direção da resultante de esforços que atua na superfície de 
ruptura é conhecida. Ela faz um ângulo φ com a normal à esta superfície.
7. monta-se um diagrama vetorial em escala e encontra-se o valor da resultante 
de empuxo ativo para esta superfície de ruptura escolhida.
8. Repete-se o processo para outras superfícies.
9. Desenha-se um gráfico relacionando os ângulos das superfícies de ruptura com 
os valores das resultantes de empuxo claculadas
10. O valor final da resultante de empuxo será o maior valor ou o pontode máximo 
da curva encontrada.
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   22
Método das cunhas ou tentativas
θ1
W
Ф
R
Ea
δ
 
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Método das cunhas ou tentativas
θ1
W
R
Ea
δ
• desenhar o vetor W em escala
Ф
• prolongamento da direção 
da resultante
• prolongamento da 
direção de Ea
• tira-se o valor de Ea
medindo o comprimento do 
vetor
 
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Método das cunhas ou tentativas
θ1
Ea
1
Ea
θ2
Ea
2
θ3
Ea
3
θ4
Ea
4
θ5
Ea
5
θ6
Ea
6
θ
• PONTO DE APLICAÇÃO E INCLINAÇÃO
• região do Ea a ser usado no projeto
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   23
• analisar também os recalques 
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   24
 
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ESCOLA DE ENGENHARIA DA UFMG 
Departamento de Engenharia de Transportes e Geotecnia 
 
  M e c â n i c a d o s S o l o s I I – P r o f . L ú c i o V i l l a r   25
 
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