Capitulo 9
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Capitulo 9


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Cap.9 \u2013 Transformação de Tensão 
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xx xy xz
yx yy yz
zx zy zz
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\uf073 \uf074 \uf073 \uf074
\uf074 \uf074 \uf073
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\uf0ea \uf0fa
\uf03d \uf0ea \uf0fa
\uf0ea \uf0fa
\uf0eb \uf0fb
\uf05b \uf05d
0
0
0 0 0
xx xy
yx yy
\uf073 \uf074
\uf073 \uf074 \uf073
\uf0e9 \uf0f9
\uf0ea \uf0fa\uf03d
\uf0ea \uf0fa
\uf0ea \uf0fa\uf0eb \uf0fb
\uf05b \uf05d xx xy
yx yy
\uf073 \uf074
\uf073
\uf074 \uf073
\uf0e9 \uf0f9
\uf03d \uf0ea \uf0fa
\uf0eb \uf0fb
= 
\uf05b \uf05d xx xy
yx yy
\uf073 \uf074
\uf073
\uf074 \uf073
\uf0e9 \uf0f9
\uf03d \uf0ea \uf0fa
\uf0eb \uf0fb
\uf05b \uf05d ' ' ' '
' ' ' '
x x x y
y x y y
\uf073 \uf074
\uf073
\uf074 \uf073
\uf0e9 \uf0f9
\uf03d \uf0ea \uf0fa
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= 
Estado de tensão em um ponto: 
Diagrama de corpo livre 
Equilíbrio de forças: 
\uf028 \uf029 \uf028 \uf029 \uf028 \uf029 \uf028 \uf029' cos cos cos sin sin cos sin sin 0x x xy xy yA A A A A\uf073 \uf073 \uf071 \uf071 \uf074 \uf071 \uf071 \uf074 \uf071 \uf071 \uf073 \uf071 \uf071\uf044 \uf02d \uf044 \uf02d \uf044 \uf02d \uf044 \uf02d \uf044 \uf03d
\uf028 \uf029 \uf028 \uf0292 2
sin 2 2sin cos ;
1 cos 2 1 cos 2
sin ; cos ;
2 2
\uf071 \uf071 \uf071
\uf071 \uf071\uf071 \uf071
\uf03d
\uf02d \uf02b
\uf03d \uf03d
' cos 2 sin 2
2 2
x y x y
x xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02b \uf02b
' 0xF \uf03d\uf0e5
Sendo que: 
2 2
' cos sin 2sin cosx x y xy\uf073 \uf073 \uf071 \uf073 \uf071 \uf074 \uf071 \uf071\uf03d \uf02b \uf02b
' 0yF \uf03d\uf0e5
\uf028 \uf029 \uf028 \uf0292 2' ' sin cos cos sinx y y x xy\uf074 \uf073 \uf073 \uf071 \uf071 \uf074 \uf071 \uf071\uf03d \uf02d \uf02b \uf02d
' ' sin 2 cos 2
2
x y
x y xy
\uf073 \uf073\uf074 \uf071 \uf074 \uf071\uf02d\uf0e6 \uf0f6\uf03d \uf02d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
Para obter a nova tensão normal na direção y\u2019 : 
90º\uf071 \uf071\uf03d \uf02b
' cos 2 sin 2
2 2
x y x y
x xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02b \uf02b
' cos 2 sin 2
2 2
x y x y
y xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02d \uf02d
2 2
' sin cos 2sin cosy x y xy\uf073 \uf073 \uf071 \uf073 \uf071 \uf074 \uf071 \uf071\uf03d \uf02b \uf02b
2 2
' cos sin 2sin cosx x y xy\uf073 \uf073 \uf071 \uf073 \uf071 \uf074 \uf071 \uf071\uf03d \uf02b \uf02b
' cos 2 sin 2
2 2
x y x y
x xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02b \uf02b
' ' sin 2 cos 2
2
x y
x y xy
\uf073 \uf073\uf074 \uf071 \uf074 \uf071\uf02d\uf0e6 \uf0f6\uf03d \uf02d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
' cos 2 sin 2
2 2
x y x y
y xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02d \uf02d
2 2
' cos sin 2sin cosx x y xy\uf073 \uf073 \uf071 \uf073 \uf071 \uf074 \uf071 \uf071\uf03d \uf02b \uf02b
\uf028 \uf029 \uf028 \uf0292 2' ' sin cos cos sinx y y x xy\uf074 \uf073 \uf073 \uf071 \uf071 \uf074 \uf071 \uf071\uf03d \uf02d \uf02b \uf02d
2 2
' sin cos 2sin cosy x y xy\uf073 \uf073 \uf071 \uf073 \uf071 \uf074 \uf071 \uf071\uf03d \uf02b \uf02d
2 2
'
2 2
'
2 2
' '
cos sin 2sin cos
sin cos 2sin cos
sin cos sin cos cos sin
x x
y y
x y xy
\uf073 \uf071 \uf071 \uf071 \uf071 \uf073
\uf073 \uf071 \uf071 \uf071 \uf071 \uf073
\uf074 \uf071 \uf071 \uf071 \uf071 \uf071 \uf071 \uf074
\uf0e9 \uf0f9\uf0ec \uf0fc \uf0ec \uf0fc
\uf0ea \uf0fa\uf0ef \uf0ef \uf0ef \uf0ef
\uf03d \uf02d\uf0ed \uf0fd \uf0ed \uf0fd\uf0ea \uf0fa
\uf0ef \uf0ef \uf0ef \uf0ef\uf0ea \uf0fa\uf02d \uf02d\uf0ee \uf0fe \uf0ee \uf0fe\uf0eb \uf0fb
\uf07b \uf07d \uf05b \uf05d\uf07b \uf07d' T\uf073 \uf073\uf03d
Exercícios: 
O estado plano de tensão em um ponto é representado pelo elemento mostrado na 
figura. Determine o estado de tensão no ponto em outro elemento orientado a 30º 
no sentido horário em relação à posição mostrada. 
Exercício: 
Tensões Principais 
\uf028 \uf029' cos 2 sin 2
2 2
x y x y
x xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02b \uf02b
Exemplo: 
80
50
25
xx
yy
xy
MPa
MPa
MPa
\uf073
\uf073
\uf074
\uf03d \uf02d
\uf03d
\uf03d \uf02d
\uf028 \uf029' ' sin 2 cos 2
2
x y
x y xy
\uf073 \uf073\uf074 \uf071 \uf071 \uf074 \uf071\uf02d\uf0e6 \uf0f6\uf03d \uf02d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
Tensões Principais 
\uf028 \uf029' cos 2 sin 2
2 2
x y x y
x xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02b \uf02b
\uf028 \uf029'
2sin 2 2cos 2 0
2
x yx
xy
\uf073 \uf073\uf073 \uf071 \uf071 \uf074 \uf071\uf071
\uf02d\uf0b6
\uf03d \uf02d \uf02b \uf03d
\uf0b6
\uf028 \uf029' cos 2 sin 2
2 2
x y x y
x xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf071 \uf071 \uf074 \uf071\uf02b \uf02d\uf03d \uf02b \uf02b
p\uf071\uf028 \uf029tan 2
2
xy
p
x y
\uf074
\uf071
\uf073 \uf073
\uf03d
\uf02d\uf0e6 \uf0f6
\uf0e7 \uf0f7
\uf0e8 \uf0f8
Tensões Principais representam a tensão normal máxima e a tensão 
normal mínima em um determinado ponto 
ângulo principal 
ou 
plano principal onde ocorrem as 
tensões máxima e mínima 
Tensões Principais 
2
2
1
2 2
x y x y
xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf074\uf02b \uf02d\uf0e6 \uf0f6\uf03d \uf02b \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
2
2
2
2 2
x y x y
xy
\uf073 \uf073 \uf073 \uf073\uf073 \uf074\uf02b \uf02d\uf0e6 \uf0f6\uf03d \uf02d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
2
x y
m
\uf073 \uf073
\uf073
\uf02b
\uf03d
onde: 
2
2
max
plano 2
x y
xy
\uf073 \uf073
\uf074 \uf074
\uf02d\uf0e6 \uf0f6
\uf03d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
Tensão principal máxima 
Tensão principal mínima 
Por convenção: 
1 2\uf073 \uf073\uf03e
\uf028 \uf029' ' sin 2 cos 2
2
x y
x y xy
\uf073 \uf073\uf074 \uf071 \uf071 \uf074 \uf071\uf02d\uf0e6 \uf0f6\uf03d \uf02d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
\uf028 \uf029' '
0
x y\uf074 \uf071
\uf071
\uf0b6
\uf03d
\uf0b6
Tensão de cisalhamento máxima: 
\uf028 \uf029
2
tan 2
x y
s
xy
\uf073 \uf073
\uf071
\uf074
\uf02d\uf0e6 \uf0f6
\uf0e7 \uf0f7
\uf0e8 \uf0f8\uf03d \uf02d s\uf071
ângulo de cisalhamento 
máximo 
2
2
max
2
x y
xy
\uf073 \uf073
\uf074 \uf074
\uf02d\uf0e6 \uf0f6
\uf03d \uf02b\uf0e7 \uf0f7
\uf0e8 \uf0f8
Exemplos: 
\u2022A transformação da tensão no plano têm uma solução gráfica que é fácil de 
lembrar. 
Círculo de Mohr \u2014 tensão no plano 
\u2022A transformação da tensão no plano têm uma solução gráfica que é fácil de 
lembrar. 
Círculo de Mohr \u2014 tensão no plano 
Exemplos: 
(1) (2) 
(3) 
\u2022A tensão de cisalhamento máxima e a tensão normal média associada podem 
também ser localizadas usando o círculo de Mohr. 
2
 
2
minmaxminmax
max abs
\uf073\uf073\uf073\uf073\uf073\uf074 \uf02b\uf03d\uf02d\uf03d avg
Tensão de cisalhamento máxima absoluta 
*Exercícios Cap 9 - Hibbeler 7ª ed 
 
10, 11, 13, 14,16, 19, 24, 26, 27, 29, 30, 31, 34, 39, 41, 
42, 43, 46, 47, 49, 76, 77, 78, 81, 82, 84, 87 
 
 
*Cap. 6 (Beer, Johnston, 3ªed) 
 
5, 6, 17, 19, 22, 23, 29, 54, 57,