MER311: Advanced Strength of MaterialsStrength of Materials

LECTURE OUTLINE

Stress Tensor Equilibrium

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 1

Stress at a PointSh i th T il (+) Di tiShown in the Tensile (+) Direction

y y

σy

y

yz yx

y

xz

xz

σ

xy

yz

zy

yx

σxσzzx

σyyx

xz σxσzz x

yx

z xxy

yz

zy

Surfaces with a PositiveDirected Area Normal

Surfaces with a NegativeDirected Area Normal

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 2

Stress Tensor

x xy xz

yx y yz

zx zy z

xx xy xz xx xy xz yx yy yz ij yx yy yz ij

zx zy zz zx zy zz

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 3

Element with Finite DimensionsElement with Finite Dimensions

y y yx

y yy y

y

yx

yx yy

yzyz y

y

σx σzxz zxFxy x

xyΔy

zy

z z

zyzy z

z

zFxF

yFx

x xx

xy xx

z

x

Δx Δzσy

yzyx

z zz

zxzx z

xzxz x

x

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MER311: Advanced Strength of Materials 4

zx z

Equilibrium EquationsSum of the Moments

xy yx y y

yz zy yz zy

xz zx

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 5

Stress Tensor yzyyx

xzxyx

x

xzxyxx

zzyzx

z

y

ij

zzzyzx

yzyyyx

xz

ijyzyyyx

xzxyxx

xy

yz

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 6

zzzyzx

Equilibrium EquationsSum of the Forces

0

xxzxyx Fzyx

0

yzyxy F

zyx

0

yF

zyx

0

zzyzxz F

zyx

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 7

yx

Example

The stress field within an elastic structural member is expressed as follows:

σx=-x3+y2, τxy=5z+2y2, τxz=xz3+x2yσy=2x3+.5y2, τyz=0, σz=4y2-z3

Determine the body force distribution required for equilibrium.

Union CollegeMechanical Engineering

MER311: Advanced Strength of Materials 8