What Is Principal Plane in Engineering Drawing

Analysis of stress and strain

JOHN CASE M.A., F.R.Ae.S , ... CARL T.F. ROSS B.Sc., Ph.D, D.Sc., C.Eng., F.R.I.North.A., M.Southward.N.A.M.E. , in Strength of Materials and Structures (Fourth Edition), 1999

five.seven Values of the principal stresses

The directions of the principal planes are given by equation (5.8). For any two-dimensional stress organisation, in which the values of σ_x, σ_y and τ_xy are known, tan2θ is calculable; 2 values of θ, separated by xc°, can and then be found. The principal stresses are so calculated by substituting these vales of θ into equation (5.6).

Alternatively, the chief stresses can be calculated more than directly without finding the principal planes. Earlier we defined a principal airplane as i on which at that place is no shearing stress; in Effigy 5.8 it is assumed that no shearing stress acts on a plane at an angle θ to Oy.

Figure five.8. A principal stress acting on an inclined plane; at that place is no shearing stress t associated with a principal stress σ.

For equilibrium of the triangular block in the 10-management,

$\sigma (ccos\theta )-{\sigma }_x(ccos\theta )={\tau }_{xy}(csin\theta )$

and so

For equilibrium of the block in the y-direction

$\sigma (csin\theta )-{\sigma }_y(csin\theta )={\tau }_{\mathrm{10}y}(ccos\theta )$

and thus

On eliminating θ between Equations (5.10) and (5.11); by multiplying these equations together, we get

This equation is quadratic in σ; the solutions are

(5.12) $\begin{array}{l}{\sigma }_1=\frac{\mathrm{one}}{2}({\sigma }_{\mathrm{ten}}+{\sigma }_y)+\frac{1}{2}\sqrt{{\left({\sigma }_x-{\sigma }_y\right)}^2+4{\tau }_{xy}^2}=\text{maximum}\text{chief}\text{stress}\\ {\sigma }_2=\frac{1}{2}({\sigma }_x+{\sigma }_y)-\frac{1}{\mathrm{two}}\sqrt{{\left({\sigma }_{\mathrm{10}}-{\sigma }_y\right)}^2+4{\tau }_{xy}^2}=\text{minimum}\text{main}\text{stress}\end{array}$

which are the values of the primary stresses; these stresses occur on mutually perpendicular planes.

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Stresses, Strains, and Elastic Response of Soils

Victor N. Kaliakin , in Soil Mechanics, 2017

4.2.5.two Determination of Principal Stress

In many cases, the maximum and minimum normal stresses associated with a given stress country, and the planes on which they deed, are of interest. To determine the orientation of the plane associated with a minimum or a maximum normal stress, Eq. (four.10) is differentiated with respect to θ and the resulting expression is set equal to zero, giving

$\frac{d{\sigma }_{{\mathrm{10}}^{\prime }}}{d\theta }=-\frac{1}{\mathrm{two}}\left({\sigma }_{\mathrm{10}}-{\sigma }_y\right)\left(2\sin 2\theta \right)-{\tau }_{\mathrm{ten}y}\left(\mathrm{ii}\cos \mathrm{two}\theta \right)=0$

Thus,

(iv.17) $\tan 2{\theta }_p=\frac{-{\tau }_{xy}}{\frac{1}{\mathrm{ii}}\left({\sigma }_x-{\sigma }_y\right)}$

where the subscript on θ indicates that this bending defines the maximum or minimum normal stress. Eq. (4.17) has two roots, since the value of the tangent of an angle in diametrically reverse quadrants is the same. These roots are thus 180   degrees apart.

The angles θ _p1 and θ _pii locate the planes on which the maximum and minimum normal stresses act, respectively. Since iiθ _pone and twoθ _pii are 180   degrees autonomously, θ _p1 and θ _ptwo will exist ninety   degrees autonomously.

These planes are called the principal planes of stress. The stresses acting on these planes—the maximum and minimum normal stresses—are called the principal stresses.

Remark: For three-dimensional stress states, the three main stresses (σ _one,σ _ii, and σ _three) are usually ordered such that σ _ane  ≥ σ ₂  ≥ σ _iii, where σ ₁ is the major principal stress, σ ₂ is the intermediate primary stress, and σ _iii is the small principal stress. There are iii principal planes that are perpendicular to each other.

Remark: The sum of the normal stresses is invariant, i.eastward., it is contained of the coordinate system. Consequently, the mean stress defined in Eq. (4.5) is expanded to

${\sigma }_{\mathrm{one\; thousand}}=\frac{1}{3}\left({\sigma }_x+{\sigma }_y+{\sigma }_z\right)=\frac{\mathrm{ane}}{3}\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)$

Effigy 4.eleven schematically illustrates the orientation of the main stress directions in σ-τ infinite. From the triangles shown in this figure,

Figure 4.11. Schematic illustration of master stress orientation.

(4.xviii) $\sin 2{\theta }_{p_1}=\frac{-{\tau }_{xy}}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{xy}\right)}^2}}$

(four.19) $\cos 2{\theta }_{p_{\mathrm{one}}}=\frac{\frac{1}{2}\left({\sigma }_{\mathrm{10}}-{\sigma }_y\right)}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{\mathrm{ten}y}\right)}^2}}$

and

(4.xx) $\sin 2{\theta }_{p_2}=\frac{{\tau }_{xy}}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{\mathrm{two}}\right)}^2+{\left({\tau }_{xy}\right)}^{\mathrm{two}}}}=-\sin 2{\theta }_{p_1}$

(iv.21) $\cos 2{\theta }_{p_2}=\frac{-\frac{\mathrm{ane}}{2}\left({\sigma }_{\mathrm{ten}}-{\sigma }_y\right)}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{\mathrm{ii}}\right)}^2+{\left({\tau }_{\mathrm{10}y}\right)}^2}}=-\cos 2{\theta }_{p_1}$

Substituting Eqs. (4.18) and (4.nineteen) into Eq. (four.10) gives the magnitude of the maximum normal stress, which corresponds to the major primary stress σ _ane, i.eastward.,

(4.22) $\begin{array}{c}{\sigma }_{x^{\prime }}=\frac{1}{2}\left({\sigma }_x+{\sigma }_y\right)+\frac{1}{2}\left({\sigma }_x-{\sigma }_y\right)\left[\frac{\frac{1}{2}\left({\sigma }_x-{\sigma }_y\right)}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{\mathrm{ten}y}\right)}^2}}\right]-{\tau }_{xy}\left[\frac{-{\tau }_{\mathrm{ten}y}}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{xy}\right)}^2}}\right]\\ =\frac{\mathrm{ane}}{\mathrm{two}}\left({\sigma }_x+{\sigma }_y\right)+\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{xy}\right)}^{\mathrm{two}}}\equiv {\sigma }_1\end{array}$

Next, substituting Eqs. (four.18) and (4.xix) into Eq. (4.16) gives the magnitude of the minimum normal stress, which corresponds to the minor principal stress σ ₂, i.due east.,

(4.23) ${\sigma }_{y^{\prime }}=\frac{1}{\mathrm{ii}}\left({\sigma }_x+{\sigma }_y\right)-\frac{\mathrm{one}}{2}\left({\sigma }_x-{\sigma }_y\right)\left[\frac{\frac{1}{2}\left({\sigma }_x-{\sigma }_y\right)}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^{\mathrm{two}}+{\left({\tau }_{\mathrm{ten}y}\right)}^2}}\right]+{\tau }_{\mathrm{10}y}\left[\frac{-{\tau }_{xy}}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{\mathrm{two}}\right)}^{\mathrm{two}}+{\left({\tau }_{xy}\right)}^{\mathrm{two}}}}\right]=\frac{\mathrm{one}}{2}\left({\sigma }_x+{\sigma }_y\right)-\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{\mathrm{10}y}\right)}^2}\equiv {\sigma }_2$

Finally, substituting Eqs. (4.eighteen) and (iv.19) into Eq. (4.13) gives

(four.24) ${\tau }_{{\mathrm{10}}^{\prime }{y}^{\prime }}=\frac{1}{2}\left({\sigma }_x-{\sigma }_y\right)\left[\frac{-{\tau }_{\mathrm{ten}y}}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{\mathrm{two}}\right)}^2+{\left({\tau }_{xy}\right)}^2}}\right]+{\tau }_{xy}\left[\frac{\frac{1}{2}\left({\sigma }_x-{\sigma }_y\right)}{\sqrt{{\left(\frac{{\sigma }_x-{\sigma }_y}{2}\right)}^2+{\left({\tau }_{\mathrm{10}y}\right)}^2}}\right]=0$

The shear stress is thus zero on the main planes of stress. ³ Figure 4.12 shows the principal stress state for atmospheric condition of plane stress.

Figure iv.12. Schematic illustration of chief stress state.

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Multiple Design Curves for Beam Lateral Buckling

N.S. Trahair , in Stability and Ductility of Steel Structures (SDSS'97), 1998

2.1 Elastic Lateral Buckling

A steel axle which is bent in its stiffer principal plane may buckle elastically in a flexural-torsional mode past deflecting u out of the airplane of loading and twisting ϕ, as shown in Figure 1. The elastic buckling resistance decreases with the beam length L, and increases with the modest axis flexural rigidity EI_y, the torsional rigidity GJ, and the warping rigidity EI_west. For a simply supported axle in uniform angle, the moment K_yz at elastic buckling (Timoshenko and Gere, 1961, Trahair and Bradford, 1991, Trahair, 1993) is given by

Figure 1. Axle elastic lateral buckling

(1) $K_{yz}={\left\{\frac{{\mathrm{\pi }}^{\mathrm{ii}}E{I}_y}{{\mathrm{50}}^2}\left(\mathrm{Chiliad}J+\frac{{\mathrm{\pi }}^2\mathrm{Due\; east}{i}_{\mathrm{due\; west}}}{L^2}\right)\right\}}^{1/\mathrm{ii}}$

Extensive enquiry has shown that the bending moment distribution has a very significant result on the elastic buckling resistance (Trahair and Bradford, 1991, Trahair, 1993), and that compatible bending is the worst case. The event of the angle moment distribution may exist immune for approximately by using

(2) ${\mathrm{Thousand}}_m={\alpha }_{\mathrm{yard}}{\mathrm{Thou}}_{yz}$

for the maximum moment at rubberband buckling, in which α_thousand is a moment modification gene. Values of α_m for many different moment distributions are available (Trahair, 1993), as well equally a general approximation (Standards Commonwealth of australia, 1990)

(3) ${\alpha }_m=\frac{\mathrm{1.seven}{M}_m}{{\left(M_1^{\mathrm{two}}+{\mathrm{Thousand}}_2^2+{\mathrm{1000}}_3^{\mathrm{ii}}\right)}^{1/2}}$

in which M ₂ , K _ane ,M ₃ are the moments at the mid- and quarter-points of the beam.

The rubberband buckling resistance of a beam is reduced by a transverse load Q which acts at a height (-y_Q) in a higher place the axis of the beam, every bit a result of boosted overturning torques generated by the load Q, the load top (-y_Q), and the twist rotation ϕ of the beam, as shown in Effigy 1c. The maximum moment at elastic buckling may frequently be approximated (Trahair, 1993) past

(four) $\frac{M_m}{{\mathrm{1000}}_{yz}}={\mathrm{\alpha }}_m\left\{\surd \left[\mathrm{i}+{\left(\frac{\mathrm{0.iv}{\mathrm{\alpha }}_{\mathrm{chiliad}}{y}_Q}{M_{yz}/P_y}\right)}^{\mathrm{two}}\right]+\frac{0.4{\mathrm{\alpha }}_m{y}_Q}{M_{yz}/P_y}\right\}$

These formulations for the elastic buckling resistance are for a only supported beam whose twist rotations ϕ are prevented at the supports, simply many practical beams may accept only elastic end restraints against twist rotation, or may take additional restraints against minor axis rotations du/dz or against warping displacements proportional to dϕ/dz. These restraints are oft deemed for by substituting an effective length

(6) $L_e=\mathit{kL}$

for the actual length L in the to a higher place formulations. Information on the effects of elastic restraints is given in Trahair (1993), and incorporated into the design procedures of Standards Australia (1990).

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Circuitous STRESSES

E.J. HEARN Ph.D., B.Sc. (Eng.) Hons., C.Eng., F.I.Mech.E., F.I.Prod.E., F.I.Diag.E. , in Mechanics of Materials 1 (Third Edition), 1997

13.5 Principal plane inclination in terms of the associated primary stress

Information technology has been stated in the previous section that expression (13.x), namely

$\tan \mathrm{ii}\theta =\frac{\mathrm{two}{\tau }_{xy}}{({\sigma }_{\mathrm{ten}}-{\sigma }_y)}$

yields ii values of θ, i.east. the inclination of the two principal planes on which the master stresses σ ₁, and σ₂ act. It is uncertain, however, which stress acts on which plane unless eqn. (13.8) is used, substituting ane value of θ obtained from eqn. (xiii.x) and observing which one of the 2 principal stresses is obtained. The post-obit culling solution is therefore to be preferred.

Consider one time over again the equilibrium of a triangular block of cloth of unit depth (Fig. thirteen.eight); this time AC is a chief plane on which a principal stress σ_p acts, and the shear stress is zero (from the property of master planes).

Fig. thirteen.8.

Resolving forces horizontally,

$\begin{array}{cc}\hfill ({\sigma }_x\times BC\times \mathrm{ane})+({\tau }_{xy}\times AB\times 1)& =({\sigma }_p\times AC\times 1)\cos \theta \hfill \\ \hfill {\sigma }_x+{\tau }_{xy}\tan \theta & ={\sigma }_p\hfill \end{array}$

Thus nosotros have an equation for the inclination of the primary planes in terms of the main stress. If, therefore, the principal stresses are determined and substituted in the above equation, each will requite the respective angle of the plane on which it acts and there tin and then exist no defoliation.

The to a higher place formula has been derived with two tensile direct stresses and a shear stress system, as shown in the figure; should any of these be reversed in action, then the appropriate minus sign must be inserted in the equation.

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31st European Symposium on Estimator Aided Process Engineering

Rachid Ouaret , ... Stéphane Negny , in Computer Aided Chemical Technology, 2021

3.2 Principal Component Analysis for interval-valued information

The interpretation of the position of the interval-valued data in the principal plane is the aforementioned as in the classical principal component analysis situation. In Effigy 2, we bear witness the results with respect to the offset three axes, accomplished by the Symbolic PCA using 4 and five(centers method). Notice that the 61% of the total inertia is explained by the get-go 2 axes in the case of simulation (40 exchangers), and 98.5\% of in the case of initial information (4 exchangers). In Effigy 2, closeness among clusters exchanger mainly influenced past the same descriptors. Merely one group consisting of exchangers 1 and 3 can be identified. The constituent elements of this cluster are influenced by the same chief factors. Exchangers ii and 4 are discrete from the cluster from a 3D perspective. This observation is in line with the conclusions of the initial study (Floquet et al., 2016). For fake information, the same assay can be accomplished using additional information, and at this phase, it is as well difficult to give an interpretation of the similarity in size and shape among exchangers. In order to shed new light on the variability of out let temperatures as a function of input streams, nosotros suggest a linear regression on interval values.

Effigy 2. Principal 3D-infinite with data of interval blazon of HEN. Factorial for 4 Exchangers (left) and for 40 Exchangers (right).

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Paraxial Rays and First-Order Optics

Rudolf Kingslake , R. Barry Johnson , in Lens Design Fundamentals (Second Edition), 2010

3.iii.i The Relation betwixt the Principal Planes

Proceeding further, we see in Figure 3.9 that a paraxial ray A traveling from left to correct is effectively bent at the 2d master plane Q and emerges through F ₂, while a similar paraxial ray B traveling from correct to left along the same straight line will be effectively bent at R and cantankerous the centrality at F ₁. Reversing the direction of the arrows along ray BRF _i yields ii paraxial rays entering from the left toward R, which become two paraxial rays leaving from the point Q to the right; thus Q is plainly an image of R, and the two principal planes are therefore conjugates. Because R and Q are at the same height to a higher place the axis, the magnification is +ane, and for this reason the principal planes are sometimes referred to as unit planes.

Effigy 3.9. The principal planes as unit of measurement planes.

When whatever arbitrary paraxial ray enters a lens from the left it is connected until it strikes the P ₁ aeroplane, and and then it jumps across the hiatus betwixt the principal planes, leaving the lens from a point on the second primary plane at the aforementioned height at which it encountered the showtime principal plane (come across Figure iii.10).

Effigy 3.10. A general paraxial ray traversing a lens.

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NATURAL CONVECTION IN AN INCLINED CYLINDER

S.S. Leong , East. Leonardi , in Transport Phenomena in Heat and Mass Transfer, 1992

7 CONCLUSIONS

Every bit the cylinder inclination γ to the gravitational vector is increased, the construction of the flow changes. The main vortices in the primary plane movement from the end walls to the middle of the cylinder before merging to class a single primary jail cell. The temperature becomes stratified in the principal aeroplane as the inclination increases and the isotherms become linear in the primal region of the cylinder. At that place is a singled-out alter in the temperature and axial velocity distribution when the inclination angle γ is greater than 140°. This is acquired past the reduced number of vortices in the central region of the cylinder. For the values studied in this investigation, the overall Nusselt number is a maximum when the cylinder is horizontal although the local Nusselt number is a maximum when the cylinder is vertical. This is due to the presence of secondary vortices which reduces the average Nusselt number.

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Multidimensional fusion by image mosaics

Yoav Y. Schechner , Shree M. Nayar , in Epitome Fusion, 2008

eight.ii.1 Background on focus

Focusing is required in well-nigh cameras. Permit the photographic camera view an object at a distance s _object from the first (forepart) master airplane of the lens. A focused epitome of this object is formed at a altitude s _epitome behind the second (back) principal aeroplane of the lens, as illustrated in Figure eight.3. For simplicity, consider first an abnormality-costless flat-field photographic camera, having an constructive focal length f. And so,

Effigy 8.3. Geometry of a uncomplicated camera organization.

(8.1) $\frac{\mathrm{ane}}{s_{\text{prototype}}}=\frac{1}{f}-\frac{1}{s_{\text{object}}}$

Hence, s _image is equivalent to s _object. The image is sensed past a detector array (e.g., a CCD), situated at a distance due south _detector from the back master aeroplane. If s _detector = s _image, then the detector array senses the focused image. Generally, however, s _detector ≠ s _prototype. If |s _image − south _detector| is sufficiently large, then the detector senses an prototype which is defocus blurred.

For a given southward _detector, there is a small range of south _paradigm values for which the defocus blur is insignificant. This range is the depth of focus. Due to the equivalence of s _prototype to s _object, this corresponds to a range of object distances which are imaged sharply on the detector array. This range is the depth of field (DOF). Hence, a single frame tin generally capture in focus objects that are in this limited span. Nevertheless, typically, unlike objects or points in the FOV have different distances s _object, extending beyond the DOF. Hence, while some objects in a frame are in focus, others are defocus blurred.

There is a common method to capture each object betoken in focus, using a stationary camera. In this method, the FOV is fixed, while K frames of the scene are acquired. In each frame, indexed k ε [1, K], the focus settings of the organisation change relative to the previous frame. Modify of the settings tin be accomplished past varying s _detector, or f, or s _object, or whatever combination of them. This mode, for any specific object point (x, y), there is a frame yard(10, y) for which Equation (8.ane) is approximated as

(8.2) $\frac{1}{{\mathrm{due\; south}}_{\text{detector}}^{\left(k\right)}}\approx \frac{\mathrm{one}}{f^{\left(\mathrm{one\; thousand}\right)}}-\frac{1}{s_{\text{object}}^{\left(k\right)}\left(\mathrm{10},y\right)}$

i.e. ${\mathrm{south}}_{\text{detector}}^{\left(k\right)}\approx {\mathrm{southward}}_{\text{image}}^{\left(m\right)}$ , bringing the image of this object point into focus. This is the focusing procedure. Since each point (x, y) is caused in focus at some frame k, and then fusing the information from all Grand frames yields an image in which all points appear in focus. This principle is sketched in Figure 8.4. The result of this image fusion is effectively a high DOF image. Even so, the FOV remains limited, since the camera is static while the frames are acquired. In the subsequent sections, nosotros will evidence that focusing and extension of the FOV can be obtained in a unmarried, efficient scan.

Figure 8.4. An image frame has a express FOV of the scene (marked by $\stackrel{\sim }{\mathrm{10}}$ ) and a limited DOF. Past fusing differently focused images, the DOF tin exist extended by image mail service processing, only the FOV remains limited.

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Applications of Rubberband Deformation

David R.H. Jones , Michael F. Ashby , in Engineering Materials 1 (5th Edition), 2019

Answers

7.1

This is a consequence of the equations of static equilibrium—the examination cube must not rotate near whatever of its axes.

7.2

Principal planes accept no components of shear stress interim on them. Principal directions are normal to chief planes. Principal stresses are tensile stresses acting on principal planes. The shear stress components all vanish.

7.iii

(a) $\left(\begin{array}{ccc}{\sigma }_1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right)$ ,

(b) $\left(\begin{array}{ccc}{\sigma }_1& 0& 0\\ 0& {\sigma }_2& 0\\ 0& 0& 0\end{array}\right)$ ,

(c) $\left(\begin{array}{ccc}–p& 0& 0\\ 0& –p& 0\\ 0& 0& –p\end{array}\right)$ ,

(d) $\left(\begin{array}{ccc}{\sigma }_{\mathrm{ane}}& 0& 0\\ 0& {\sigma }_{\mathrm{ii}}& 0\\ 0& 0& {\sigma }_3\end{array}\right)$ .

7.4

$\left(\begin{array}{ccc}0& \tau & 0\\ \tau & 0& 0\\ 0& 0& 0\end{array}\right)$ .

7.5

$\left(\begin{array}{ccc}{\sigma }_{\mathrm{one}}& 0& 0\\ 0& 0.5{\sigma }_{\mathrm{one}}& 0\\ 0& 0& 0\end{array}\right)$ . At that place are no shear stress components normal to these axes.

7.6

Considering the two shear strain terms on any given axis airplane are defined, at that place is no rotation.

seven.vii

Principal strains are tensile strains. The shear strain components all vanish.

7.eight

(a) $\left(\begin{array}{ccc}{\varepsilon }_1& 0& 0\\ 0& –{\mathit{\upsilon \varepsilon }}_{\mathrm{i}}& 0\\ 0& 0& –{\mathit{\upsilon \varepsilon }}_{\mathrm{i}}\end{array}\right)$ .

(b) $\left(\begin{array}{ccc}\varepsilon & 0& 0\\ 0& \varepsilon & 0\\ 0& 0& \varepsilon \end{array}\right)$ .

7.nine

Compare Figure 3.5(b) with Effigy vii.five.

$\begin{array}{l}{\mathrm{eastward}}_{13}=\gamma \text{and}{e}_{31}=0.\\ {\varepsilon }_{13}={\varepsilon }_{31}=\frac{1}{2}\left({\mathrm{east}}_{\mathrm{thirteen}}+e_{31}\right)=\frac{\gamma }{\mathrm{ii}}.\end{array}$

vii.ten

From Figure 3.5(c), the dilatation is defined every bit $\mathit{\Delta }=\frac{\mathrm{\Delta }5}{5}.$

Consider a cube of material of unit side. For each of the three principal (axial) strains, the increase in the volume of the cube is equal to the strain. For example, a strain ɛ ₁ makes the cube longer in the ane direction by ɛ _one. The increase in the volume of the cube is ɛ ₁  ×   one   ×   1   = ɛ _ane. The dilatation produced past ɛ _ane is therefore (ɛ _i  ×   i   ×   1)/(i   ×   1   ×   one)   = ɛ ₁. Therefore, Δ  = ɛ _ane  + ɛ ₂  + ɛ ₃.

7.xi

Take a cube of the textile having its faces normal to the principal directions, and employ a stress σ ₁ along the ane direction. Since $E=\frac{{\sigma }_1}{{\varepsilon }_{\mathrm{i}}},{\varepsilon }_{\mathrm{ane}}=\frac{{\sigma }_{\mathrm{one}}}{\mathrm{Eastward}}$ . Now apply a stress σ ₂ along the 2 direction. This will produce a strain forth the ii direction of ${\varepsilon }_2=\frac{{\sigma }_2}{E}$ . In plow, this strain will produce a strain forth the i direction of ${\varepsilon }_1=–{\mathit{\upsilon \varepsilon }}_2=–\upsilon \frac{{\sigma }_{\mathrm{two}}}{E}$ . The net strain along the 1 management at present becomes ${\varepsilon }_1=\frac{{\sigma }_{\mathrm{one}}}{\mathrm{Eastward}}–\upsilon \frac{{\sigma }_2}{E}$ . Side by side apply a stress σ ₃ forth the 3 direction, and repeat as before to find the total net value of ɛ ₁, equally given by the first equation. Repeat this process to find the equations for ɛ _two and ɛ _iii. Sum the iii primary strains using the three equations. It is then straightforward to show that

$\mathit{\Delta }=\frac{\left(\mathrm{i}–2\upsilon \right)}{\mathrm{Due\; east}}\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right).$

The dilatation is zippo when υ  =   1/2.

7.12

For uniaxial tension, $\mathit{\Delta }=\frac{\left(\mathrm{one}–\mathrm{ii}\upsilon \right)}{\mathrm{Due\; east}}{\sigma }_1=\left(1–2\upsilon \right){\varepsilon }_1.$

The volume changes are therefore 0.4ɛ _ane, ɛ ₁, and 0.

vii.13

From Equation (iii.8), $K=–\frac{p}{\mathit{\Delta }}$ . For hydrostatic pressure loading, the equation

$\mathit{\Delta }=\frac{\left(\mathrm{i}–2\upsilon \right)}{E}\left({\sigma }_{\mathrm{i}}+{\sigma }_{\mathrm{ii}}+{\sigma }_3\right)$ becomes $\mathit{\Delta }=–\frac{\left(1–2\upsilon \right)}{\mathrm{Eastward}}\left(3p\right)$ . Then $K=–p\times \frac{E}{–\left(\mathrm{one}–\mathrm{two}\upsilon \right)3p}=\frac{E}{3\left(1–2\upsilon \right)}.$ This shows that Thou  = E when υ  =   1/three.

7.fourteen

The solid safe sole is very resistant to being compressed, because it is restrained against lateral Poisson's ratio expansion by beingness glued to the relatively stiff sole. Nevertheless, the molded surface has a much lower resistance to existence compressed, considering the lateral Poisson's ratio expansion of each carve up rubber cube tin occur without constraint (provided the gaps between adjacent cubes do non shut upwardly completely). So your colleague is right.

seven.15

The axial forcefulness practical to the cork to push button information technology into the bottle results in a null lateral Poisson's ratio expansion, and then information technology does not become any harder to push the cork into the neck of the bottle. However, the axial force applied to the rubber hurl results in a large lateral Poisson's ratio expansion, which makes it almost incommunicable to force the bung into the neck of the bottle.

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Deformation in the context of energy geostructures

Lyesse Laloui , Alessandro F. Rotta Loria , in Analysis and Design of Free energy Geostructures, 2020

4.5.three Principal stresses

A feature of the stress tensor, similar to the strain tensor, is that at every fabric point of any coordinate organization there be iii mutually perpendicular planes, called chief planes, along which zero shear stresses are observed. The normal stresses on these planes are chosen main stresses.

The stress vector acting on a principal aeroplane is characterised by only the normal component. Therefore considering Eq. (4.x) and by indicating with ${\mathrm{due\; north}}_i$ the unit vector of a principal plane characterised by direction cosines $n_{\mathrm{ten}},n_y$ and $n_z$ in a rectangular Cartesian coordinate arrangement $x,y,z$ , and with ${\lambda }_{*}$ the modulus of the respective component, the stress vector can exist expressed as

(4.17) ${\sigma }_{\mathit{ji}}{\mathrm{due\; north}}_j={\lambda }_{*}{\mathrm{northward}}_i$

Eq. (4.17) is equivalent to the following eigenvalue problem

(iv.xviii) $\left({\sigma }_{\mathit{ij}}-{\lambda }_{*}{\delta }_{\mathit{ij}}\right)n_j=0$

that in extended form reads

(four.19) $\{\begin{array}{c}{(\sigma }_{\mathit{xx}}-{\lambda }_{*}{)n}_x+{\sigma }_{\mathit{xy}}{\mathrm{northward}}_y+{\sigma }_{\mathit{xz}}{n}_z=0\\ {\sigma }_{\mathit{xy}}{n}_x+{(\sigma }_{\mathit{yy}}-{\lambda }_{*})n_y+{\sigma }_{\mathit{yz}}{n}_z=0\\ {\sigma }_{\mathit{xz}}{\mathrm{north}}_{\mathrm{10}}+{\sigma }_{\mathit{zy}}{n}_y+{(\sigma }_{\mathit{zz}}-{\lambda }_{*})n_z=0\end{array}$

with

(4.20) $n_x^{\mathrm{two}}+{\mathrm{northward}}_y^{\mathrm{ii}}+n_z^{\mathrm{two}}=1$

Eqs (4.19) and (4.xx) lead to a nontrivial solution for the direction cosines only if the determinant of the coefficients is zip. This condition leads to the characteristic cubic equation (also called characteristic polynomial)

(4.21) ${{\lambda }_{*}}^{\mathrm{iii}}-I_{\mathrm{ane}}{{\lambda }_{*}}^2+I_2{\lambda }_{*}-I_{\mathrm{three}}=0$

where $I_1,I_2$ and $I_{\mathrm{iii}}$ are 3 coefficients that exercise not modify for unlike coordinate transformations, that is they are invariants. Whatever linear combination of invariants still is an invariant.

The solution of the higher up equation is found for 3 real eigenvalues ${\lambda }_{*\mathrm{i}}={\sigma }_1,{\lambda }_{*\mathrm{ii}}={\sigma }_2$ and ${\lambda }_{*\mathrm{iii}}={\sigma }_3$ that correspond the primary stresses. Differently to the components of the stress tensor that modify for dissimilar coordinate systems taken with reference to the same point, the three principal stresses are invariants nether coordinate transformation. The commutation of each of the iii eigenvalues in Eq. (4.nineteen) allows calculating the eigenvectors $n_{\mathrm{one}},{\mathrm{due\; north}}_{\mathrm{ii}}$ and ${\mathrm{northward}}_{\mathrm{three}}$ that represent the principal directions.

The three invariant coefficients expressed in Eq. (iv.21) are the start, second and 3rd stress invariants (or invariants of the stress tensor) and are given by

(4.22) $\{\begin{array}{c}{I}_1=\mathrm{tr}\left({\sigma }_{\mathit{ij}}\right)={\sigma }_{\mathit{ii}}={\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}+{\sigma }_{\mathit{zz}}\\ I_2=\frac{1}{\mathrm{two}}\left({\sigma }_{\mathit{ii}}{\sigma }_{\mathit{jj}}-{\sigma }_{\mathit{ij}}{\sigma }_{\mathit{ij}}\right)={\sigma }_{\mathit{xx}}{\sigma }_{\mathit{yy}}+{\sigma }_{\mathit{yy}}{\sigma }_{\mathit{zz}}+{\sigma }_{\mathit{zz}}{\sigma }_{\mathit{twenty}}-{\sigma }_{\mathit{xy}}^2-{\sigma }_{\mathit{yz}}^2-{\sigma }_{\mathit{zx}}^{\mathrm{two}}\\ I_{\mathrm{iii}}=\det {\sigma }_{\mathit{ij}}={\sigma }_{\mathit{xx}}{\sigma }_{\mathit{yy}}{\sigma }_{\mathit{zz}}+2{\sigma }_{\mathit{xy}}{\sigma }_{\mathit{yz}}{\sigma }_{\mathit{zx}}-{\sigma }_{\mathit{20}}{\sigma }_{\mathit{yz}}^2-{\sigma }_{\mathit{yy}}{\sigma }_{\mathit{zx}}^{\mathrm{ii}}-{\sigma }_{\mathit{zz}}{\sigma }_{\mathit{xy}}^2\end{array}$

Past setting equal to cipher the off-diagonal components of the stress tensor in Eq. (four.22), the relations between the invariants and the principal stresses tin can be institute.

Culling formulations of the stress invariants can be derived from the stress tensor itself instead from the feature polynomial and read

(four.23) $\{\begin{array}{c}{I}_{1*}=\mathrm{tr}\left({\sigma }_{\mathit{ij}}\right)={\sigma }_{\mathit{2}}=I_{\mathrm{i}}\\ I_{2*}=\frac{\mathrm{one}}{2}{\sigma }_{\mathit{ij}}{\sigma }_{\mathit{ij}}=\frac{1}{2}{I}_1^{\mathrm{ii}}-I_2\\ I_{\mathrm{iii}*}=\frac{1}{3}{\sigma }_{\mathit{ik}}{\sigma }_{\mathit{km}}{\sigma }_{\mathit{mi}}=\frac{\mathrm{one}}{3}{I}_1^3-I_{\mathrm{ane}}{I}_{\mathrm{ii}}+I_3\end{array}$

Invariants tin also be expressed in similar forms for the deviatoric stress tensor, for example, equally

(4.24) $\{\begin{array}{c}{J}_1=\mathrm{tr}\left({\mathrm{southward}}_{\mathit{ij}}\right)={\mathrm{south}}_{\mathit{two}}={\sigma }_{\mathit{ii}}-{\sigma }_{\mathit{nn}}=0\\ J_2=\frac{\mathrm{ane}}{\mathrm{two}}{\mathrm{southward}}_{\mathit{ij}}{s}_{\mathit{ij}}=\frac{1}{2}\left[{\left({\sigma }_{\mathit{xx}}-p\right)}^{\mathrm{two}}+{\left({\sigma }_{\mathit{yy}}-p\right)}^{\mathrm{ii}}+{\left({\sigma }_{\mathit{zz}}-p\right)}^{\mathrm{ii}}+2s_{\mathit{xy}}^2+2{\mathrm{south}}_{\mathit{yz}}^2+2s_{\mathit{xz}}^2\right]\\ =I_{\mathrm{ii}*}-\frac{I_{1*}^2}{\mathrm{vi}}\\ J_3=\frac{1}{3}{s}_{\mathit{ik}}{\mathrm{southward}}_{\mathit{km}}{s}_{\mathit{mi}}=I_{3*}-\frac{2}{3}{I}_{\mathrm{i}*}{I}_{2*}+\frac{2}{27}{I}_{1*}^{\mathrm{three}}\end{array}$

When a coordinate system is called such that the directions are parallel to the principal directions, the stress tensor reduces to

(4.25) ${\sigma }_{\mathit{ij}}=\left[\begin{array}{ccc}{\sigma }_{\mathrm{i}}& 0& 0\\ 0& {\sigma }_{\mathrm{two}}& 0\\ 0& 0& {\sigma }_{\mathrm{three}}\end{array}\right]$

It is often convenient to number the principal stresses (without mandatorily referring to the position of the stress components in the stress tensor reported in Eq. iv.26) so that

(4.26) ${\sigma }_1\ge {\sigma }_2\ge {\sigma }_{\mathrm{iii}}$

The principal directions are mutually orthogonal because the eigenvectors of a symmetric tensor, such as the stress tensor, are mutually orthogonal. Eqs (4.17)–(4.26) written thus far for the stress tensor tin also be written for the strain tensor.

In two dimensions, analogous results can be obtained with Eq. (4.21) that reduces to

(4.27) ${{\lambda }_{*}}^2-\left({\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}\right){\lambda }_{*}+\left({\sigma }_{\mathit{xx}}{\sigma }_{\mathit{yy}}-{\sigma }_{\mathit{xy}}^2\right)=0$

The two master stresses in the airplane may then be written explicitly equally

(4.28) ${\lambda }_{*}={\sigma }_{\mathrm{1,2}}=\frac{{\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}}{2}\pm \sqrt{{\left(\frac{{{\sigma }_{\mathit{20}}-\sigma }_{\mathit{yy}}}{2}\right)}^2+{\sigma }_{\mathit{xy}}^2}$

with the chief directions that make an angle ${\phi }_x$ with the $\mathrm{10}$ -axis of

(iv.29) $\tan {\phi }_x=\frac{n_y}{n_x}=\frac{{\sigma }_{\mathit{xy}}}{{\sigma }_{\mathit{xx}}-{\lambda }_{*}}$

In two dimensions, the coordinate points ( ${\sigma }_{\mathrm{one}};0)$ and ( ${\sigma }_2;0)$ represent peculiar points of the so-called Mohr circle of stress [for farther details, run into, eastward.1000. Timoshenko (1953)], which is given by the post-obit equation

(4.30) ${\sigma }_{{\mathrm{ten}}_1{y}_1}^2+{\left({\sigma }_{x_{\mathrm{one}}{x}_1}-\frac{{\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}}{\mathrm{ii}}\right)}^2={\sigma }_{\mathit{xy}}^2+\frac{\mathrm{ane}}{4}{\left({\sigma }_{\mathit{xx}}-{\sigma }_{\mathit{yy}}\right)}^{\mathrm{ii}}$

where ${\sigma }_{x_1{x}_{\mathrm{one}}}$ and ${\sigma }_{x_1{y}_1}$ are the normal and shear stress components interim in whatever management $x_1,y_1$ , where the $x_1$ centrality makes an bending ${\alpha }_{{\mathrm{10}}_1}$ with the $\mathit{x}$ axis (positive in the anticlockwise management).

The Mohr circle of stress represents the setting of all possible stress states acting on a point along dissimilar planes (cf. Fig. 4.8). The formulas for the stress components derived from the Mohr circle of stress are

Figure 4.8. A typical Mohr circle.

(four.31) $\{\begin{array}{c}{\sigma }_{{\mathrm{ten}}_1{x}_1}=\frac{\left({\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}\right)}{\mathrm{two}}+\frac{\left({\sigma }_{\mathit{twenty}}-{\sigma }_{\mathit{yy}}\right)}{\mathrm{ii}}\cos 2{\alpha }_{x_1}+{\sigma }_{\mathit{xy}}\sin {2\alpha }_{x_{\mathrm{i}}}\\ {\sigma }_{y_1{y}_1}=\frac{\left({\sigma }_{\mathit{xx}}+{\sigma }_{\mathit{yy}}\right)}{\mathrm{two}}-\frac{\left({\sigma }_{\mathit{xx}}-{\sigma }_{\mathit{yy}}\right)}{2}\cos \mathrm{two}{\alpha }_{x_1}-{\sigma }_{\mathit{xy}}\sin {\mathrm{two}\alpha }_{{\mathrm{ten}}_1}\\ {\sigma }_{x_1{y}_{\mathrm{ane}}}=-\frac{\left({\sigma }_{\mathit{xx}}-{\sigma }_{\mathit{yy}}\right)}{\mathrm{two}}\sin 2{\alpha }_{{\mathrm{10}}_1}+{\sigma }_{\mathit{xy}}\cos {2\alpha }_{x_{\mathrm{i}}}\end{array}$

The use of invariants defined in Eqs (4.22)–(four.24) is often practical for a number of considerations related to constitutive modelling of geomaterials and their interaction with geostructures. In particular, invariants of physical involvement especially in the framework of plasticity theory are $I_{\mathrm{one}}$ , $J_2$ and $J_3$ . Physically, $I_{\mathrm{one}}$ represents the magnitude of the mean stress, $J_2$ represents the magnitude of the deviatoric stress and $J_{\mathrm{iii}}$ determines the direction of the deviatoric stress (Yu, 2006). The same concepts tin be analysed graphically in the Haigh–Westergaard infinite, that is the three-dimensional space where the main directions of stresses are selected as coordinate axes (cf. Fig. 4.9). The value of $I_1$ provides a measure of the distance along the space diagonal ( ${\sigma }_1={\sigma }_{\mathrm{two}}={\sigma }_3=I_{\mathrm{one}}/3$ ) from the origin to the current spherical aeroplane, too called octahedral aeroplane, divers as

Figure 4.9. Representations in the Haigh–Westergaard space: definition of key variables in (A) the π-plane and octahedral plane and (B) the octahedral airplane but.

(4.32) ${\sigma }_1+{\sigma }_2+{\sigma }_{\mathrm{three}}=I_1=\mathrm{three}p$

This distance reads

(4.33) $\overline{\mathit{AA}\prime }=\frac{\left({\sigma }_1+{\sigma }_2+{\sigma }_{\mathrm{three}}\right)}{\sqrt{3}}=\frac{\sqrt{3}}{\mathrm{iii}}{I}_1=\sqrt{3}p$

The special plane for which the mean stress $p$ is zero is chosen the π-plane and reads

(4.34) ${\sigma }_{\mathrm{one}}+{\sigma }_2+{\sigma }_3=0$

The second invariant of the deviatoric stress tensor $J_{\mathrm{ii}}$ is a mensurate of the distance from the space diagonal to the current stress country in the spherical plane. The combination of $J_2$ and $J_3$ through the formulation of the Lode'southward bending defines the orientation of the stress country within this plane and reads

(iv.35) ${\theta }_l=-\frac{\mathrm{one}}{\mathrm{iii}}{\sin }^{-1}\left(\frac{\mathrm{three}\sqrt{3}}{2}\frac{J_{\mathrm{three}}}{J_{\mathrm{ii}}^3}\right)$

The 2d invariant of the deviatoric stress tensor $J_2$ is also related to the deviatoric stress as (Roscoe and Burland, 1968; Wood, 1990)

(4.36) $q=\sqrt{3J_{\mathrm{ii}}}$

that in terms of principal stresses tin can also be written every bit

(4.37) $q=\frac{\mathrm{one}}{\mathrm{six}}\left[{\left({\sigma }_1-{\sigma }_{\mathrm{ii}}\right)}^{\mathrm{ii}}+{\left({\sigma }_2-{\sigma }_3\right)}^2+{\left({\sigma }_1-{\sigma }_3\right)}^2\right]$

In Fig. four.9, the principal stresses ${\tilde{\sigma }}_1$ , ${\tilde{\sigma }}_2$ and ${\tilde{\sigma }}_3$ are the projections in space of the principal stresses ${\sigma }_1$ , ${\sigma }_2$ and ${\sigma }_3$ .

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