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User:Eml4500.f08.team.guan/hw5

Derivation of Applying the P.V.W to Eliminate Rows Corresponding to the Boundary Conditions in the Stiffness Matrix
In this lecture, the justification of eliminating rows 1, 2, 5, and 6 in the global stiffness matrix of the 2-bar truss problem in order to obtain the 2 by 2 stiffness matrix $$\bar{k}_{2x2}$$ will be shown. The derivation is as follows. Recall the global force displacement relationship. $$\displaystyle K_{6x6}d_{6x1}=F_{6x1}$$ Subtracting the force column matrix to both sides will yield the following equation. $$\displaystyle K_{6x6}d_{6x1}-F_{6x1}=0_{6x1}...(1)$$ where 0 is a 6 by 6 column matrix of zeros. Now, the principle of virtual work is applied. The principle of virtual work states that "forces applied to a static system do no virtual work". Also, for rigid bodies, if "a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero; and conversely, if the total virtual work of all external forces acting on a rigid body is zero then the body is in equilibrium" (source: ). Applying it, we have the following equation. $$w_{6x1} \cdot (Kd-F)_{6x1}=0_{1x1}...(2)$$ Equation 1 is true for all matrices $$w_{6x1}$$. Here, a virtual weighting function ("$$w$$") or coefficient is dotted with KD-F. The weighting function or coefficient means relative weight between functions. In equation 2, 0 is a 1 by 1 matrix meaning it is a scalar. In order to multiply two column matrices and result in scalar, the dot product must be used. Now, the fact that equation 1 is equivalent to equation 2 is trivial, however, the fact that equation 2 is equivalent to equation 1 is not trivial. Here, we have $$w \cdot (Kd-F)=0$$ for all $$w$$. To prove this, let us select some choices for the weighting function matrix. Choice 1: We select the weighting function such that, $$\displaystyle w_1=1, w_2=0, w_3=0, w_4=0, w_5=0, w_6=0$$ $$w^T= \begin{bmatrix} 1 & 0& 0& 0& 0& 0\\ \end{bmatrix} $$

Plugging this column matrix into equation 2 will yield the following. $$w \cdot (Kd-F)= 1[\sum_{j=1}^6{K_{1j}d_j-F_1}]+0[\sum_{j=1}^6{K_{2j}d_j-F_2}]+0[\sum_{j=1}^6{K_{3j}d_j-F_3}]+...+0[\sum_{j=1}^6{K_{6j}d_j-F_6}]=0 $$ Multiplying it all out gives $$\sum_{j=1}^6{K_{1j}d_j}=F_1 $$

For choice 2, we set the second row of the weighting column matrix to be 1 and the reset of the values to be 0. $$\displaystyle w_1=0, w_2=1, w_3=0, w_4=0, w_5=0, w_6=0$$ $$w^T= \begin{bmatrix} 0 & 1& 0& 0& 0& 0\\ \end{bmatrix} $$

Plugging this column matrix into equation 2 will yield the following. $$w \cdot (Kd-F)= 0[\sum_{j=1}^6{K_{1j}d_j-F_1}]+1[\sum_{j=1}^6{K_{2j}d_j-F_2}]+0[\sum_{j=1}^6{K_{3j}d_j-F_3}]+...+0[\sum_{j=1}^6{K_{6j}d_j-F_6}]=0 $$ Multiplying it all out gives $$\sum_{j=1}^6{K_{2j}d_j}=F_2 $$

Considering all the choices of the weighting function column matrix, it is concluded that $$\displaystyle Kd=F$$ Thus, equation 1 is confirmed. However, this did not eliminate rows 1, 2, 5, and 6. The boundary conditions for the 2-bar truss problem must also be adhered to while considering the principle of virtual work. Here are the boundary conditions. $$\displaystyle d_1=d_2=d_5=d_6=0$$ The weighting column matrix (now called the virtual displacement) must be "kinematically admissible" meaning that it also must follow the boundary conditions. This means that $$\displaystyle w_1=w_2=w_5=w_6=0$$ From 10.1, ou can see that $$ Kd$$ can be reduced to a rectangular matrix multiplied by a 2 by 1 matrix and ultimately, after applying boundary conditions and the principle of virtual work with corresponding virtual displacements equaling zero, we have the following expression. $$ w \cdot (Kd-F)= \begin{bmatrix} w_3\\ w_4\\ \end{bmatrix}\cdot

(\bar{K} \bar{d}-\bar{F})=0...(3) $$ for all $$w_3$$ and $$ w_4$$ where $$ \bar{K}=\begin{bmatrix} k_{33} & k_{34}\\ k_{43} & k_{44}\\ \end{bmatrix} , \bar{d}= \begin{bmatrix} d_{3}\\ d_{4} \\ \end{bmatrix} , \bar{F}=\begin{bmatrix} F_{3}\\ F_{4} \\ \end{bmatrix} $$

title: User:Eml4500.f08.team.guan/hw5


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User:Jenny O/Final thoughts

What have we explored and learnt?
Social psychology is far more interesting and much broader than I ever imagined. In my experience it has drawn together a range of topics and has put them into perspective. I often found myself returning to textbooks and notes from other subjects to re-consider what I had taken away at the time. For that reason alone I think it is very well paced as a final third year unit. An interesting observation I have made in this degree is that studying psychology enables you to articulate what we have already learned or intuitively know about yourself and others. This has certainly been the case in social psychology. Having said that, my intuition and experiences are not always correct. The degree has certainly heightened my self-awareness, and my awareness of others. It would be interesting to hear how others feel about it.

We have covered a wide range of very interesting topics in this unit. As I have said however, I thought the way in which the social self was presented in the textbook was poorly managed and very confusing. That was unfortunate as I believe this is the most fundamental concept in social psychology (Fiske, 2004 did a much better job). The work we did on prejudice, aggression, and prosocial behaviour and applying these to the Rwandan Genocide and the work of Jane Elliott was very interesting and enabled us to gain a much deeper understanding of the concepts involved. A highlight for me was the application of social psychology to our immediate social and cultural world (the Australian Zeitgeist). I am now a fan of the Australian social researcher, Hugh McKay, who really struck a chord with me. Now that I have some time I intend to explore his work. Another surprising highlight was Environmental Psychology (Maybe because it is so fresh in my mind). Almost overnight, I have developed an unprecedented level of environmental concern, mainly due to the credible way in which the information was covered. (As we now know this concern is an important precursor to behavioural change). I think I have subconsciously ignored many environmental issues because of the alarmist way they are presented to us in the media – after a while you become habituated to their tactics. The application of psychology to the built environment also fascinated me. The notion that psychology could play such an important role just didn’t occur to me - I always assumed that was the job of a good architect (and they are few and far between in suburban Australia). We could have covered more on culture and maybe moved away from the pervasive, egocentric, American approach to psychology.

Is it worth knowing?
Yes, absolutely, all of this is worth knowing but we all have to do something with it. Applying it is a crucial part of the process. Understanding social psychology plays an important role in developing a greater awareness of yourself and others. If more people developed this understanding and applied it appropriately there might be less turmoil in the world.

How have we been limited by theory and method?
Unfortunately, much of this content is underpinned by a very ethnocentric, American zeitgeist, which, as I have said, fails to consider a large portion of humanity [According to Rozin (2001, 2007) somewhere between 90 and 99%]. This was something addressed in my essay. This narrow view is a major flaw in social psychology. I was astounded by the patronising and paternalistic tone of the textbook, which added to my distrust of the 'all-American-way'. I think there are more balanced and more interesting textbooks (e.g., Fiske, 2004 or Vaughan & Hogg, 2005). However, I have also thought to myself, maybe that is exactly what it is - social psychology (and much of psychology in general) is about white, middle-class, Westerners and should not be construed as anything else. If that’s the case, it should be made clear at the outset.

How worthwhile has it been?
This has been an incredibly worthwhile experience, and a great way to finish the degree. Thanks James.

Alternative textbooks
Fiske, S.T. (2004). Social beings: A core motives approach to social psychology. Hoboken, NJ: John Wiley.

Vaughan, G. M. & Hogg, M. A. (2005). Introduction to social psychology. Frenchs Forest, NSW: Pearson Education.

title: User:Jenny O/Final thoughts


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User:Eml4500.f08.jamama.justin/HW5

$$\hat{W}$$ = virtual axial displacement

W = Virtual displacement in global coordinate system corresponds to $$d^{(e)}$$ (4X1)

recall last equations:

$$q^{(e)}=T^{(e)}d^{(e)}$$  (3)

$$ \hat{w}=T^{(e)}w $$   (4)

Replace equations (3) and (4) into equation (2). (From page meeting 26, page 2)

$$(T^{(e)}w)\bullet \left[\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-P^{(e)} \right]=0$$ for all $$\hat{w}$$ (4X1) (5)

Recall: $$\left(AB \right)^{T}=B^{T}A^{T}$$ (6)

pXqqXr

Recall:$$a\bullet b=a^{T}b$$ (7)

nx1nx1(1xnnx1)

(1x1 scalar)

Apply (7) and (6) in (5):

$$(T^{(e)}w)^{T}\bullet \left[\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-P^{(e)} \right]=0$$

$$(T^{(e)T}w^{T})\bullet \left[\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-P^{(e)} \right]=0$$

$$\Rightarrow w\bullet \left[\left( T^{(e)}\hat{k}^{(e)}T^{(e)}\right)d^{(e)}-T^{(e)T}P^{(e)} \right]=0$$ for all w (4x1)

$$\Rightarrow w\bullet \left[k^{(e)}d^{(e)}-f^{(e)} \right]= 0 $$ for all w.  $$\Rightarrow k^{(e)}d^{(e)}=f^{(e)}$$

So far, discrete case (non-continuous) matricies. Now, continuous case (Partial Differential Equations) motivational model problem: Elastic bar with varying A(x), E(x), subject to varying axial loads (distributed) + concentrated load + inertia force (dynamic). Axial loads being the dependent.

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User:Eas4200c.f08/Plate buckling

Rectangular plate under in-plane distributed load

Simply-supported boundary conditions
Refs: MIT OCW 2.081J / 16.230J Plates and Shells Spring 2007 Lecture notes, pp.54-61. Sun, Mechanics of Aircraft Structures, Wiley, 2006, Section 7.7. Timoshenko & Gere, Theory of elastic stability, McGraw-Hill, 1961, Sec.9.2, p.351.

For a simply-supported rectangular plate under in-plane distributed compressive load, see figure Lecture notes, p.57. Plate dimensions: Length $$\displaystyle a$$ along $$\displaystyle x$$ axis, length $$\displaystyle b$$ along $$\displaystyle y$$ axis, thickness $$\displaystyle h$$. Young's modulus $$\displaystyle E$$, Poisson's ratio $$\displaystyle \nu$$. Total resultant compressive load $$\displaystyle P_x$$ (force) pointed in the $$\displaystyle x$$ direction, distributed along the $$\displaystyle y$$ axis.

Buckling mode shapes $$\displaystyle \psi_{mn} (x,y)$$
 * {| style="width:100%" border="0"

$$  \displaystyle \psi (x,y) =   c_{mn} \sin \left(     \frac      {m \pi x}      {a}   \right) \sin \left(     \frac      {n \pi y}      {b}   \right) \, \ {\rm for} \ m,n = 1, 2, 3, \ldots $$ where $$\displaystyle c_{mn}$$ is a constant, $$\displaystyle (m,n)$$ are the number of half wave-lengths along $$(x,y)\;$$, respectively. The transverse displacement (out of plane) $$\displaystyle u_z (x,y)$$ is a linear combination of the buckling mode shapes:
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (1a)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle u_z (x,y) =  \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \psi_{mn} (x,y) $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (1b)
 * }

Critical buckling load $$\displaystyle (P_x)_{cr}$$

 * {| style="width:100%" border="0"

$$  \displaystyle (P_x)_{cr} =  k_c \frac {\pi ^2 D}  {b} $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (2)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle k_c (m, a/b) :=  \left(      \frac      {m b}      {a}      +      \frac      {a}      {m b}   \right)^2 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (3)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle D  := \frac {E h^3} {12 (1 - \nu^2)} $$ The factor $$\displaystyle D$$ is called the plate bending stiffness, and is the equivalent of the bending stiffness $$\displaystyle EI$$ in beam bending. Note that the dimension of the second area moment of inertia $$\displaystyle I$$, denoted by $$\displaystyle [I]$$, is $$\displaystyle L^4$$, i.e., length to the power 4:
 * style="width:95%" |
 * style="width:95%" |
 * style= | (4)
 * }

\displaystyle [I] = L^4 $$ Since $$\displaystyle \sigma = E \epsilon$$, we have

\displaystyle [\sigma] =  [E] [\epsilon] =  [E] =  \frac {F} {L^2} $$ i.e., force per length squared, as $$\displaystyle [\epsilon] = 1$$. It follows that the dimension of the beam bending stiffness coefficient

\displaystyle [EI] =   \frac {F} {L^2} L^4 =  FL^2 $$ whereas the dimension of plate bending stiffness coefficient $$\displaystyle D$$ is

\displaystyle [D] =  [E] [h^3] =  \frac {F} {L^2} L^3 =  FL $$ since the dimension of the Poisson's ratio is $$\displaystyle [\nu] = 1$$.

In addition, with $$\displaystyle [k_c] = 1$$, we have

\displaystyle [(P_x)_{cr}] =  [k_c] \frac {[D]} {[b]} =  1   \cdot \frac {FL} {L} =  F $$ which is consistent with the definition of $$\displaystyle P_x$$, the total in-plane compressive load.

Critical buckling stress $$\displaystyle (\sigma_{xx})_{cr}$$

 * {| style="width:100%" border="0"

$$  \displaystyle (\sigma_{xx})_{cr} =  \frac {(P_x)_{cr}} {bh} =  k_c \frac {\pi ^2 D}  {b^2 h} $$
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (7)
 * }

Since the factor $$\displaystyle k_c$$ depends on the number of half wave-lengths $$\displaystyle m$$ and the aspect ratio $$\displaystyle a/b$$, for a given aspect ratio $$\displaystyle a/b$$ of the rectangular plate, the minimum critical load $$\displaystyle (\sigma_{xx})_{cr}^\star$$ and the associated number of half wave-lengths $$\displaystyle m^\star$$ are obtained by minimizing the function $$\displaystyle k_c$$, i.e.,
 * {| style="width:100%" border="0"

$$  \displaystyle (\sigma_{xx})_{cr}^\star =  k_c^\star \frac {\pi ^2 D}  {b^2 h} $$ with
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (8)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle k_c^\star (a/b) =  \min_{m = 1, 2, \cdots} \ k_c (m, a/b) =  \min_{m = 1, 2, \cdots} \left(     \frac      {m b}      {a}      +      \frac      {a}      {m b}   \right)^2 =  \left(      \frac      {m^\star b}      {a}      +      \frac      {a}      {m^\star b}   \right)^2 $$ where
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (9)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle m^\star (a/b) =  \underset{m=1,2, \cdots}{\rm argmin} \ k_c (m, a/b) =  \underset{m=1,2, \cdots}{\rm argmin} \left(     \frac      {m b}      {a}      +      \frac      {a}      {m b}   \right)^2 $$ i.e., $$\displaystyle m^\star (a/b)$$ is the minimizer of $$\displaystyle k_c (m,a/b)$$ for a given aspect ratio $$\displaystyle a/b$$.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (10)
 * }

For a simply-supported column, see Lecture notes, p.77 and Sun [2006, Sec 7.2], the critical load is

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$$  \displaystyle (P_{cr})_{column} =  \frac {\pi^2 EI} {L^2} $$ where $$\displaystyle L$$ is the length of the column.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (11)
 * }

See also Lecture notes, pp.4-5, University of Auckland, New Zealand.

Clamped boundary conditions
For rectangular plates with clamped boundary conditions on all 4 edges under in-plane distributed load, use the Excel file for calculating critical stress: Be sure to save the excel file you downloaded somewhere since you may not be able to download it again within one day.

The above excel file is based on the table in the book by Young & Budynas, Roark's formula for stress and strain, McGraw-Hill, 7th edition, 2002, p.730, where the following formula and notation were used
 * {| style="width:100%" border="0"

$$  \displaystyle \sigma^\prime =  K   \frac{E}{1 - \nu^2} \left(     \frac{t}{b}   \right)^2 $$ where $$\displaystyle \sigma^\prime \equiv (\sigma_{xx})_{cr}$$, $$\displaystyle K$$ is a function different from $$\displaystyle k_c$$, and $$\displaystyle t \equiv h$$ the plate thickness.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (12)
 * }

For simply-supported plates, comparing Eq.(12) to Eq.(7), one can deduce that
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$$  \displaystyle K   = k_c \frac{\pi^2}{12} =   \left(      \frac      {m b}      {a}      +      \frac      {a}      {m b}   \right)^2 \frac{\pi^2}{12} $$ For clamped plates, we still have $$\displaystyle K = k_c \pi^2 / 12$$, but the coefficient $$\displaystyle k_c$$ is no longer given by Eq.(3).
 * style="width:95%" |
 * style="width:95%" |
 * style= | (13)
 * }

When the clamped plate is close to being square, the following (buckling) shape $$\displaystyle \psi (x,y)$$ can be used to estimate the buckling load with reasonable accuracy (Timoshenko & Gere [1961], Theory of elastic stability, McGraw-Hill, p.386.)

 * {| style="width:100%" border="0"

$$  \displaystyle \psi (x,y) =  \frac {c} {4}  \left(      1 - \cos \frac{2 \pi x}{a}   \right) \left(     1 - \cos \frac{2 \pi y}{b}   \right) $$
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 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (14)
 * }

Another calculator of critical stress (free for square plate), Poisson's ratio cannot be changed, not useful.

Simply-supported boundary conditions
Refs: Timoshenko & Gere, Theory of elastic stability, McGraw-Hill, 1961, Sec.9.7, p.379. Sun, Mechanics of Aircraft Structures, Wiley, 2006, Section 7.7.4.

Because of the simply-supported conditions, the buckling mode shapes $$\displaystyle \psi (x,y)$$ are given as in Eq.(1a).

Denote the shear force per unit length of the plate as $$\displaystyle N_{xy}$$, i.e.,
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$$  \displaystyle N_{xy} =  \sigma_{xy} h $$ where $$\displaystyle h$$ is the plate thickness. See Fig.7.23 on p.263 in Sun [2006]. In the skin or spar webs of an airfoil section, $$\displaystyle N_{xy}$$ is the same as the shear flow $$\displaystyle q$$.
 * style="width:95%" |
 * style="width:95%" |
 * style= | (20)
 * }

Stored strain energy in the plate
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$$  \displaystyle U  = \frac{D}{2} \frac{\pi^4 a b}{4} \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} C_{mn}^2 \left(     \frac{m^2}{a^2}      +      \frac{n^2}{b^2}   \right)^2 $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (21)
 * }

Work done by the external shear force per unit length $$\displaystyle N_{xy}$$
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$$  \displaystyle W  = - 4 N_{xy} \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \sum_{p=1}^{\infty} \sum_{q=1}^{\infty} C_{mn} C_{pq} \frac {mnpq} {(m^2 - p^2)(n^2 - q^2)} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (22)
 * }

Equating the stored strain energy to the work done by external shear load, i.e.,

\displaystyle U = W $$ we obtains the critical buckling shear load $$\displaystyle (N_{xy})_{cr}$$
 * {| style="width:100%" border="0"

$$  \displaystyle (N_{xy})_{cr} =  -   \frac{ab D}{32} \frac {     \displaystyle \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} C_{mn}^2 \left(        \frac{m^2 \pi^2}{a^2}	 +         \frac{n^2 \pi^2}{b^2}      \right)^2 }  {      \displaystyle \sum_{m=1}^{\infty} \sum_{n=1}^{\infty} \sum_{p=1}^{\infty} \sum_{q=1}^{\infty} C_{mn} C_{pq} \frac{mnpq}{(m^2 - p^2)(n^2 - q^2)} } $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (23)
 * }

The next step is to find the coefficients $$\displaystyle C_{mn}$$ and $$\displaystyle C_{pq}$$ that minimize $$\displaystyle (N_{xy})_{cr}$$ to give $$\displaystyle (N_{xy})_{cr}^\star$$ i.e., the minimum critical load. To do so, take the derivatives of $$\displaystyle (N_{xy})_{cr}$$ with respect to each coefficient $$\displaystyle C_{mn}$$ and set these derivatives to zero; we obtain a set of homogeneous equations with zero right-hand side in terms of the unknown coefficients $$\displaystyle C_{mn}$$.

Introduce the following definitions:
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$$  \displaystyle \vartheta :=  \frac{a}{b} $$ which is the aspect ratio of the plate, and
 * style="width:95%" |
 * style="width:95%" |
 * style= | (24)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle \lambda :=  -   \frac{\pi^2}{32 \vartheta} \frac{\pi^2 D}{b^2 h (\sigma_{xy})_{cr}} $$
 * style="width:95%" |
 * style="width:95%" |
 * style= | (25)
 * }

The first 5 of these equations read as follows
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$$  \displaystyle \left[ \begin{array}{lllll} \frac{\lambda (1 + \vartheta^2)^2}{\vartheta^2} &	 \frac{4}{9} &	 0	 &	 0	 &	 0	 \\	 \frac{4}{9} &	 \frac{16 \lambda (1 + \vartheta^2)^2}{\vartheta^2} &	 - \frac{4}{5} &	 - \frac{4}{5} &	 \frac{36}{25} \\	 0	 &	 - \frac{4}{5} &	 \frac{\lambda (1 + 9 \vartheta^2)^2}{\vartheta^2} &	 0	 &	 0	 \\	 0	 &	 - \frac{4}{5} &	 0	 &	 \frac{\lambda (9 + \vartheta^2)^2}{\vartheta^2} &	 0	 \\	 0	 &	 \frac{36}{25} &	 0	 &	 0	 &	 \frac{\lambda (9 + 9 \vartheta^2)^2}{\vartheta^2} \end{array} \right] \left\{ \begin{array}{l} C_{11} \\	 C_{22} \\	 C_{13} \\	 C_{31} \\	 C_{33} \end{array} \right\} =  \left\{ \begin{array}{l} 0	 \\	 0	 \\	 0	 \\	 0	 \\	 0     \end{array} \right\} $$ Let $$\displaystyle \mathbf K _{5 \times 5}$$ be the 5x5 matrix shown in Eq.(26). Notice the nice symmetry in $$\displaystyle \mathbf K _{5 \times 5}$$: (1) It is symmetric, (2) On the diagonal coefficients, there is a factor 9 that appears with the index 3 in the coefficients $$\displaystyle C_{mn}$$, e.g., compare ($$\displaystyle K_{33}$$ for $$\displaystyle C_{13}$$) with ($$\displaystyle K_{44}$$ for $$\displaystyle C_{31}$$), and with ($$\displaystyle K_{55}$$ for $$\displaystyle C_{33}$$).
 * style="width:95%" |
 * style="width:95%" |
 * style= | (26)
 * }

Case 1: Truncate the above system in Eq.(26) so to use only the first two equations in Eq.(26) with only two unknowns $$\displaystyle C_{11}$$ and $$\displaystyle C_{22}$$, i.e.,
 * {| style="width:100%" border="0"

$$  \displaystyle \left[ \begin{array}{ll} \frac{\lambda (1 + \vartheta^2)^2}{\vartheta^2} &	 \frac{4}{9} \\	 \frac{4}{9} &	 \frac{16 \lambda (1 + \vartheta^2)^2}{\vartheta^2} \end{array} \right] \left\{ \begin{array}{l} C_{11} \\	 C_{22} \end{array} \right\} =  \left\{ \begin{array}{l} 0	 \\	 0     \end{array} \right\} $$ with the matrix now denoted by $$\displaystyle \mathbf K _{2 \times 2}$$. Set the determinant of $$\displaystyle \mathbf K _{2 \times 2}$$ to zero, i.e.,
 * style="width:95%" |
 * style="width:95%" |
 * style= | (27)
 * }

\displaystyle \det \mathbf K _{2 \times 2} =  0 $$ one finds
 * {| style="width:100%" border="0"

$$  \displaystyle \lambda =  \pm \frac{1}{9} \frac{\vartheta^2}{(1 + \vartheta^2)^2} $$ and thus by Eq.(25), the minimum critical buckling shear stress is
 * style="width:95%" |
 * style="width:95%" |
 * style= | (28)
 * }
 * {| style="width:100%" border="0"

$$  \displaystyle (\sigma_{xy})_{cr}^{\star,2} =  \pm k_c^{\star,2} \frac{\pi^2 D}{b^2 h}  \, \ k_c^{\star,2} =  \frac{\pi^2}{32} \frac{9 (1 + \vartheta^2)^2}{\vartheta^3} $$ where the superscript 2 in $$\displaystyle (\sigma_{xy})_{cr}^{\star,2}$$ simply means that this minimum critical buckling stress was obtained with a 2x2 system, and the $$\displaystyle \pm$$ simply means that the critical buckling load does not depend on the direction of the shear stress (unlike the compressive buckling case). Compare Eq.(29) for $$\displaystyle (\sigma_{xy})_{cr}^{\star,2}$$ to Eq.(8) for $$\displaystyle (\sigma_{xx})_{cr}^{\star}$$.
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style="width:92%; padding:10px; border:2px solid #8888aa" |
 * style= | (29)
 * }

It was found that using a 2x2 system was not accurate enough; for a square plate, there was a 15% error for $$\displaystyle (\sigma_{xy})_{cr}^{\star,2}$$ when compared to a more accurate estimate when more unknowns were used. This error increases with for increasing aspect ratio $$\displaystyle \vartheta = a/b$$.

Case 2: Now use all 5 equations in Eq.(26) with all 5 unknowns $$\displaystyle \{ C_{11}, C_{22}, C_{13}, C_{31}, C_{33} \}$$. Setting the determinant of $$\displaystyle \mathbf K _{5 \times 5}$$ to zero, i.e.,

\displaystyle \det \mathbf K _{5 \times 5} =  0 $$ one finds
 * {| style="width:100%" border="0"

$$  \displaystyle \lambda =  \left[ \frac{\vartheta^4}{81 (1 + \vartheta^2)^4} \left\{ 1 	 + 	 \frac{81}{625} +	 \frac{81}{25} \left(	   \frac{1 + \vartheta^2}{1 + 9 \vartheta^2}	 \right)^2 +	 \frac{81}{25} \left(	   \frac{1 + \vartheta^2}{9 + \vartheta^2}	 \right)^2 \right\} \right]^{1/2} $$ Using the definition of $$\displaystyle \lambda$$ in Eq.(25), we obtain the minimum critical buckling stress
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 * }
 * {| style="width:100%" border="0"

$$  \displaystyle (\sigma_{xy})_{cr}^{\star,5} =  \pm k_c^{\star,5} \frac{\pi^2 D}{b^2 h}  \, \ k_c^{\star,5} =  \frac{\pi^2}{32} \frac{1}{\lambda \vartheta} $$
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 * style= | (31)
 * }

Clamped boundary conditions
Use a numerical method such as the finite element method: I am looking for eager students who want to do a Highest-Honors thesis.

Refs: Young & Budynas, Roark's formula for stress and strain, McGraw-Hill, 7th edition, 2002, p.732 (a table with values of $$\displaystyle k_c$$ for $$\displaystyle \vartheta = 1, 2, \infty$$ and a reference to a 1931 technical report). Szilard, Theory and analysis of plates, Prentice Hall, New Jersey, 1974, p.704 (a graph and a reference to a 1964 book and a 1960 report).

 * {| style="width:100%" border="0"

$$  \displaystyle (\sigma_{xy})_{cr} =  K   \frac {E} {1 - \nu^2} \left(     \frac{h}{b}   \right)^2 $$
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 * }

It is also indicated in Young & Budynas [2002] that experimental results yielded a value of $$\displaystyle K = 4.1$$ for large $$\displaystyle \vartheta = a/b$$. The lower experimental value for $$\displaystyle K$$ is likely due to imperfections in the experimental plate specimens.

title: User:Eas4200c.f08/Plate buckling


text:
User:Triplesugariron

 * UNAM
 * Faculty of Medicine
 * Second year Student

 * WikiUnam

Finding the best way to improve education, self-education and helping the community to look for new perspectives in any way possible.

Lecture 2.0

Medical School Curriculum

title: User:Triplesugariron


text:
Help:Headings

{| border="1" cellpadding="2" cellspacing="0" !What it looks like !What you type Section headings

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Subsection

Using more equals signs creates a subsection.

A smaller subsection

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Section headings
Headings organize your writing into sections. The Wiki software can automatically generate a table of contents from them.

Subsection
Using more equals signs creates a subsection.

A smaller subsection
Don't skip levels, like from two to four equals signs. Start with two equals signs; don't use single equals signs.
 * }

title: Help:Headings


text:
University of Florida/Eml4500/f08

 * A physics class at the University in Florida.
 * Student assignments involving using MATLAB
 * Instructor and main page: Dr Vu-Quoc User:Eml4500.f08
 * There were over 200 students in the class.

title: University of Florida/Eml4500/f08


text:
User talk:Triplesugariron

 Hello Triplesugariron, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

 * Take a guided tour and learn to edit.
 * Visit a (kind of) random random project.
 * Browse everything on Wikiversity, or visit a portal corresponding to your educational level: pre-school, primary, secondary, tertiary, non-formal education.
 * Find out about research activities on Wikiversity.

 * Read an introduction for teachers and find out how to write an educational resource for Wikiversity.
 * Give feedback about your initial observations
 * Discuss Wikiversity issues or ask questions at the colloquium.
 * Chat with other Wikiversitans on #wikiversity-en.

And don't forget to explore Wikiversity with the links to your left. Be bold to contribute and to experiment with the sandbox or your userpage, and see you around Wikiversity! Jtneill - Talk - c 03:19, 31 October 2008 (UTC)

 Hi! Our name is Vicky-Verity. We are twins and are Wikiversity mascots. We think Wikiversity is a great place to learn and make friends (when we aren't sleeping). Leave us a message!

Medicine
Check out Category:Medicine. Maybe you can help? -- Jtneill - Talk - c 03:19, 31 October 2008 (UTC)

title: User talk:Triplesugariron


text:
Fiction writing/VirileMail

VirileMail is a science fiction novel written at the Fiction Wikia. The audio resource you will find here is equivalent to this page: VirileMail:How the story was created. It is basically a conversation about how a science fiction novel was written in collaboration, using IRC. The 2 people who are talking are JWSchmidt (John W. Schmidt) and Davichito (David). John wrote most parts of the story and David helped resume the story (he wrote 4 chapters).

Audio resource

 * [[Image:VirileMail.m4a.OGG|VirileMail]]
 * play time: 17 minutes 33 seconds

Questions
These are the questions asked by David to John, in the audio resource.

1) Who was the first writer of the story? What was the original title?

2) What was the story initially about? Where did the "eggplant" come from?

3) Why did you decide to make it into a science fiction story?

4) Did you get to a writer's block before I came in? Why?

5) Where did you learn about the science (nanotechnology) for making the story?

6) How was the title selected?

John asks David:

7) How did you find VirileMail?

Answers
1) The person who started the story edited without registering a user account. The IP address seems to be assigned to the USA department of transportation, so it is fun to imagine some bored government employee starting a story about a guy driving to work at a "top speed of 135". The original title was "We need a new title" (later shortened to "WNANT").

2) As created by the first editor, it was just a short description of a guy with a boring job that involved monitoring some computer network. After a few weeks, another person who edited from an IP from Canada added some more content to the story, creating a strange event that disrupted the operation of the computer network. A few days later I noticed the story and continued with more development of the "strange event" for the computer network, asking the question: what had mysteriously disrupted the computer network?. The story was becoming a mystery. A few weeks later, Eric119 added a strange experience for a computer programmer in the story, the so called "Attack of the Eggplant" and the main character was given a last name, "Joe", became "Joe Daonet". So by that point in the story's development, the Joe Daonet was transformed from a bored employee into the hero of the story who was faced with the mystery of trying to figure out a series of strange events that had started to take place around him.

3) When I started contributing to "VirileMail", I was writing a science fiction story that is set in the "Exodemic Universe", which involves the idea that aliens from distant planets have been visiting Earth for many millions of years...without revealing themselves to the people on Earth. I started struggling to make a coherent story that incorporated the mysterious "eggplant" that Eric119 had written into the story and I finally proposed that the eggplant be a vehicle for a tiny artificial life form. So it took about two months before the story was designated as science fiction. A week later the second chapter held a description of the "VirileMail" software development project. The story was growing more complex and we faced the problem of how to efficiently discuss development of the plot. Until then, we were just using talk pages, then the "meta level" discussions for how to develop the story were moved to a special "meta page". The goal was to have different pages for readers to discuss the story and for writers who were creating the question. Also, a cover page was made with an illustration of the "eggplant". One of the fun aspects of creating fiction in a wiki is that illustrations can be easily added. Development of the story continued off-and-on for over a year, but then mostly came to a stop.

4) I had gotten busy at Wikiversity and had little time for writing fiction. A user talk page question in 2008 brought me back to the story. In my reply to that question, I tried to explain how I had gotten "stuck" while trying to develop the plot of the story: the basic problem was that some contributors seemed to be just adding random content to the story. That approach had been the basis for getting the story started, but once the plot was established, additional random additions were too difficult to incorporate into the story. The solution was to remove some of the suggested content that did not really fit into the story and then move on.

5) The idea of thinking about something that is the size of an eggplant as being the equivalent of a spaceship or an airplane required imagining a small life form that could fit into such a small vehicle. I got interested in nanotechnology when I read Engines of Creation by Eric Drexler. Drexler was one of the first people to popularize the idea of nano-scale machines. As a biologist I'm interested in the microscopic components of biological organisms and the idea that artificial life might be possible on an even smaller, nanoscale. Biological organisms might not be optimized for size: it might be possible to make smaller artificial life forms, which would have implications for space travel. Why send large biological organisms between stars when you can send smaller nanoscale artificial life forms?

6) An important part of how the story was finally completed involved making use of IRC (internet relay chat) for discussions of plot development. We also spent a significant amount of time trying to think of a good title since the original title was just "We need a new title". We wanted a balance between having a title that was clearly relevant to the story while not giving away a key aspect of the plot. Eventually we adopted "VirileMail" which is a funny name for a software development project that is explained in the story as arising from a translation error. Another way we used IRC was to compose part of the story by having a conversation in an IRC chat channel. The log of the IRC chat was incorporated directly into the story.

David answers

7) VirileMail was one of the first stories I found at Fiction wikia. As I am very interested on science fiction (I am a fan of Isaac Asimov) I started browsing that section in the Archives. I got attracted to the title (We Need A New Title) and I started to read the story. I liked it even more when I knew it was about computers; it is a topic I am very interested. After that, I told my mother to read the story; she also likes science fiction. We both were amazed that such great novels could be in a free wikia and then she read the story to the end. It was incomplete at that time so she said to me that I should try to contact the author of the story, for him to complete it. She thought the story was so good that it was a must to finish it. So, I asked the sysops who the author was and I found JWSchmidt. He was not active those days and had abandoned the story. So, I told him to complete the story and he answered me the most strange thing in the world: He told me that he could only finish the story if I could help writing it. I was amazed, since I am no native English speaker, but I agreed, the story was worth the effort. I had had a hard work trying to start "The search for Kalid", my first English story, but I thought I would give it a try. Before those stories, all of my works were translations from my Spanish ones (I am a native Spanish speaker). So, Kalid and VirileMail helped me improve my written English. And it was very interesting and fulfilling for me. I learned a lot about writing fiction, helped by John Schmidt, which is a very good writer, in my opinion.

Links

 * Fiction writing
 * Collaborative writing

title: Fiction writing/VirileMail


text:
User:Sandy~enwikiversity/Week 2 The Social Self

Lecture 2 – Notes

 * Culture – collective storehouse of information and knowledge, cultural practices we pass on to our young.
 * Social Brain Theory''' – understand our brains in part by our development of social complexities and complex culture.
 * Psyche – the mind. A psychological combination of biological inherit and what we’ve done with that culture.
 * Nature – physical world.
 * Evolution – We have got to this point through a long history of survival through breading. Random change that may evolve.
 * Natural Selection – survival (until reproduction)

Tutorial
Definition of social psychology -The collective mind A - Affect B - Behaviour C - Cognition

 * Individuals make up society, experiences, nature, nurture
 * A combination of environment and society changes us
 * What you want to know
 * Behaviour control
 * The self – looking glass theory
 * Social influence and behaviour

Readings

The Social Self
The self comes in to being at the junction between the biological processes of the inner body and the socio-cultural network to which we belong and its store of common beliefs and practices. Even without meeting other people we would still have a concept of self as a body separate to the external environment.

The self is an important tool we use to make our way through society and thereby satisfy our needs. To be effective at this the human self has taken shape through deep powerful drives such as being concerned about how others perceive us. Not only does this drive serve the goal of survival and reproduction but as social beings we care generally what others think of us. As cultural beings we have self awareness, or self concepts, which enable us to develop elaborate sets of beliefs about ourselves.

The interpersonal self or public self is part of the self that helps us to connect socially to each other. Most of us convey a certain image to others. This differs from self-conceptin as much as you may not believe the image you are portraying to be true. Negative traits like annoying habits can be left out.

The agent self or executive function is the part that gets things done, makes choices and exerts self control.

Self as impulse refers to a person’s inner thoughts and feelings as opposed to self as institution which is the way we behave in public; particularly the various roles we play in society.

Culture and Interdependence
Whether we are in a dependant or interdependent culture dictates what the self is, how we see ourselves and what motivates the self to accomplish.
 * Interdependent self-concept or self-construal is what makes you different from others.
 * Dependent self-construal is what connects you to others/groups.

Social roles
We are not designed to be by ourselves and as social beings the purpose of the self is to gain social cceptance. We have the ability to learn and adapt to behave properly and to conform to social rules and norms which in turn improves our chances of appeal and acceptance by others.

Erving Goffman analysed most human behaviour and selfhood in terms of actors playing roles.

Cultures are large systems within which we need to find one or several roles that culture defines such as a doctor or lawyer. Most of these roles are ways of relating to each other in a cultural system. Without society the self would not exist in full and without self-awareness and self-hood self-knowledge and the capacity to turn our attention back on to our source would not be possible. One of the things that sets us apart from other animals is our ability to change roles though out a life time.

Self-Awareness
Self-awareness is directing our attention at the self. It is vital for self regulation which is how we control and change ourselves. It usually involves evaluating ourselves against set standards and as a result coming up with good or bad evaluations about ourselves. These standards are ideas that have the power to cause and shape behaviour. Increased self-awareness makes our attitudes more consistent. Self-awareness enables us to behave better according to cultural standards which supports the idea that inner processes serve interpersonal functions to help us get along with others. Duval and Wicklund’s (1972) self awareness theory argues a state of self-awareness is unpleasant. Comparing ourselves to ‘average people’ or people not doing as well as us is healthier for our self-awareness than comparing ourselves to unrealistic standards such as super models. We have one of two reactions when we don’t meet those standards. We either try to fix the problem by improving ourselves or escaping from the problem by avoiding or reducing our self-awareness and therefore reducing those negative feelings (i.e. through alcohol use or suicide). Simply changing the standard rather than ourselves is often easier.

Self-awareness is vital for self regulation which is the process we use to control and change ourselves, our thoughts feelings and behaviours.

Public self-awareness is the how we are perceived by others. Thinking about ourselves through other peoples eyes. Looking outwardly to understand the self.

Private self-awareness attends to our inner needs such as emotions, thoughts desires and traits. Looking inwardly to understand the self.

Self Knowledge
Charles Horton Cooley (1902) – Looking glass self - we learn about ourselves from how we think we appear to others. The three components to this theory are:
 * We imagine how we appear to others
 * We imagine how others will judge us
 * We develop a positive or negative emotional response to this judgement

George Herbert Mead (1934) – added to Cooley’s theory that by suggesting most self-knowledge comes from feedback from others about ourselves telling us who and what we are. He called them the generalised other. The limitations of these theories stems from the fact that the feedback we get from others about ourselves is often not very accurate, honest or even consistent. We may also disregard the feedback depending on the incoming information and the source. As a result our self-knowledge often doesn’t match our public perception.

 * Introspection – the process by which we look inwardly to examine our mind and mental states. Privileged access refers to the fact that only I have direct access to my mind and feelings and therefore the only one who can truly know oneself. This also has its limits. Up to the age of eleven children think their parents know them better than they do. Nesbitt and Wilson (1977) also argued that people don’t really have privileged access to know how their mind works and guess or make assumptions as to what they assume are socially desirable answers when explaining their actions. We know what we think and feel but we don’t always know why.

Social Comparison
Festinger (1954) - social comparison theory – suggests people learn about themselves through comparing themselves to others. Self-Perception and the Overjustification Effect Bem’s (1965) - self-perception theory - self knowledge comes from learning about ourselves through observation.
 * Upward social comparison – comparing yourself with others better than you. This can inspire you to do better.
 * Downward social comparison – comparing yourself with worse off than yourself can make you feel good.
 * Intrinsic motivation – performing an activity for the sake of it.
 * Extrinsic motivation – performing an activity for its result or reward.

Overjustification Effect – The self perception theory predicts extrinsic motivation will win over intrinsic motivation as a result of the end reward and that rewards turn play in to work. Once a reward (i.e. you are paid) is offered and subsequently expected (extrinsic motivation) for an activity it is difficult to resume the activity for the sake of it (intrinsic motivation). If the reward isn’t expected it doesn’t diminish intrinsic motivation (Lepper et al.,1973).

The Fluctuating Image(s) of Self - The phenomenal self or the working self concept is what you see of yourself at any one time or the image of self that is active in our thoughts. Different aspects of us stand out at any one time and come to the forefront of our knowledge into the phenomenal self i.e. awareness that you are the only person in a group with dark or white skin.

Motivations for wanting Self-Knowledge
We seek self knowledge in many and varied ways such as undertaking personality tests. Self knowledge has many advantages including social acceptance. It is important to know our strengths and weaknesses to help us fit in with others or groups. The three motivations to seek self knowledge:
 * Appraisal motive – the desire to know the truth about ourselves including our abilities.
 * Self-enhancement motive – the desire to know favourable things about ourselves, this includes disregarding any criticism while exaggerating good qualities.
 * Consistency motive (self-verification motive) – the desire to verify our self-concepts through feedback that confirms what we already know (or think we know) about ourselves.

The self-enhancement is the strongest of the three motives for self-knowledge because we like to hear good things about ourselves. This motive is particularly important to us as it has an emotional appeal and when positive feedback is received it indicates we are accepted by others. The second strongest motive is the consistency motive; we like to have what we already believe about ourselves confirmed. This motive has a cognitive appeal confirming the way we think. As is the case with the appraisal motive it is important, even though sometimes difficult, for us to know the truth about ourselves as a basis for decisions but it is even more important to us to be shown signs we are accepted by others and to receive confirmation of our existing beliefs about ourselves.

The Duplex Mind
In different ways different parts of the duplex mind cultivates our self-knowledge. It is relevant to the interplay between the conflicting motives. The automatic systems tends to favour the self-enhancement motive as we like to think everything that’s good is because of me but everything that’s bad isn’t the result of me (something else is to blame). This automatic egotism emerges when we are under stress or preoccupied. Our conscious system is then needed to put a balanced and consistent view of the self and a more accurate appraisal of the facts.

Self Handicapping
Putting obstacles in the way of achieving is known as self handicapping. Self handicapping has a two-fold purpose. It can be used as an excuse for under achieving or used to boost our reputation. For example getting drunk the night before an exam gives us an excuse for failure if we don’t do well or boosts our reputation of being intelligent if a high grade is achieved even though we hadn’t studied and got drunk instead.

The self reference effect is any information that has an impact or relates to the self is retained better than any other information.

The endowment effect is similar in as much as items that belong to are of greater value. An item that is connected to the self becomes more valuable (in monetary and sentimental terms). Choosing an item is of greater value than being given something.

Can Self-Concept Change?
Our identity does change but very slowly. There are several theories supporting revising self-knowledge. We can consciously decide to change the way we think about ourselves and our actions will follow to reflect the new you or we can decide to change our behaviour which can subsequently change our concept will follow.

Our social world carries powerful expectations for the self to remain constant. We make the assumption that people remain the same over lengths of time which to a certain degree is true. Our social environment changing is the most common cause for the self-concept to change. If we remain in the same social circle our self-esteems tends to stay the same and even when we gradually change that same social circle tends not to notice. If we move to a new social circle then the new version of us apparent.

The looking class theory
Suggests we know ourselves through what we think other people’s opinions of us are. Therefore if those ‘others’ change (new social circle) then how we think they see us is also likely to change. Inner processes are also tied to interpersonal relations so if our social circle changes and we interact with new people our inner self is also likely to change. This supports the theory that our inner self is there to support interpersonal processes and our inner self needs to adapt to changing interpersonal processes.

On the other side of the coin, for us to change it is important for us to use our social environment. For example if we decide to get fit we join a gym or boot-camp with trainers and other people with the same goal. If you persuade others that you have changes and they accept and expect you to behave differently it is easier to maintain the changed behaviour. Self and identity require social validation. Once we have revised our inner self it is likely we will then revise our stories and memories about our lives to suit our new self.

Self-Esteem
Self-esteem is how favourably we evaluate ourselves. If your self-esteem is high you have a positive evaluation of yourself. If your self-esteem is low it isn’t that you have a negative evaluation about yourself but that the strong positive views just don’t exist. There has been much research about low self-esteem resulting in the following conclusions:
 * People with low self-esteem do not want to fail and are often less confident that they will achieve their goal.
 * Ideas about themselves are often confusing and uncertain with views about themselves to be contradictory.
 * Rather than focusing on self-enhancement their focus is on self-protection to avoid loss of esteem. They tend not to take chances or pursue opportunities to avoid, failure, embarrassment and rejection.
 * They are more likely to be emotional with more emotional highs and lows which include mood swings and overreactions than those with high self-esteem.

Choosing our comparison group carefully helps us to gain and retain self-esteem. Comparing ourselves to someone not quite as good as us makes us feel good about ourselves. The two main benefits of high self-esteem are initiative and it simply feels good. People with high self-esteem are more confident to speak up in groups, approach people and make friends and resist influence. Making us feel good provides a store of good feelings that we can draw upon when things don’t go well or as planned. It helps us overcome bad feelings. People with high self-esteem are less likely to give up and will try harder.

On the down-side people with high self-esteem are more likely to be adventurous with drugs, sex and other temptations. Another negative aspect related to high self-esteem is narcissism which is excessive self love and self focus. Narcissists believe they are better than others and believe they should be treated better than others. High self-esteem has also been associated with higher prejudice believing there group is better than other groups and discriminating more heavily in favour of their group.

People with high self-esteem continue to pursue self-esteem by dominating others and to increase their competencies whereas those with low self-esteem pursue it by seeking validation from others and avoiding failures. The pursuit of self-esteem weakens our intrinsic motivation as we are pressured to live up to other’s expectations and may exhibit behaviours others approve of for social acceptance.

Self-Deception Strategies
Self-deception strategies are the cognitive tricks we use to sustain the positive illusions we have about ourselves. One self-deception theory is the self-serving bias which is the ability to take credit for our achievements but not accept blame for our failures blaming external factors (see self-handicapping). Another related strategy is our ability to more sceptical of bad feedback than good feedback. Greenwald’s (1988) junk mail theory of self-deception suggests we have the ability to quickly identify and disregard bad or unwelcome news diminishing its impact and making it easier to forget. We also have the ability to misconstrue our view of other people to convince ourselves that our good traits are unusual but our bad traits are common.

Sociometer Theory
The sociometer theory suggests self-esteem is our measure of how our socially accepted we think we are. This can explain why we are so concerned with self-esteem as it is very strongly linked to our social acceptance.

Self-Presentation
As cultural animals we care what other people think. Self presentation is any behaviour that conveys an image of ourselves or some information about ourselves to others. Self presentation is not only how we dress, it is from how we speak about ourselves to what car we drive to the way we behave including hiding emotions so others think positive thoughts about us. Making a good impression and maintaining a good reputation through good self-presentation are important aspects of obtaining social acceptance. Good self-presentation involves showing ourselves to have good traits and not bad ones. The limitations of good self-presentation lie with the audience you are presenting to. If the values of the audience differ from ours we are likely to conform to and behave in a manner we know will meet the audience’s expectations and approval. This is usually the case but on the other hand we may deliberately present ourselves in a manner we know the audience will not approve of. Either way it may mean we are not presenting our true self.

We aspire to have many identities, which may range from a university student, to daughter, to an athlete or defending a certain value. To claim theses identities we need social validation. There is a trade-off between favourability and feasibility. We like to present ourselves in the best light but if we think we might be found out our claims are exaggerated we will tone them down. Good self-presentation helps us to create a good first impression. When an interpersonal relationship is developed this good self-presentation is less important and modesty becomes more natural and common among friends. One of the reasons for this is our friends are often familiar with our faults and failures and are likely to know when we are exaggerating. Modesty enables us to get along with each other more harmoniously.

Self-presentation is often stronger than self preservation and may lead to risky behaviours such as smoking to portray what we might think is a cool adult like image. Taking such risks indicates we value the gain of social acceptance more than our health and it is more of a motivation than the motivation to stay alive and healthy. This supports the theory that our psyche is designed to seek social acceptance. We are also more concerned with our self-presentation than our private self-esteem.

Skinnerian behaviourism – The theory that emphasised learning from reward and punishment as the main cause for behaviour.

Culture is a network of meaning and as such what we do depends partially on what those meanings are. Meaning connects things giving actions meanings when they are connected to other things or events. We use thought to imagine our actions before we physically carry them out and most actions can be described as meaning different things.

Levels of Meaning
Actions are made up of high and low levels of meaning which range from the more complex and longer time span (bigger picture) to the smaller here and now detail. When we experience unpleasantness and the bigger picture becomes overwhelming we like to focus on the lower level meanings. Focusing on the lower level also helps us to solve problems. In others words if we are trying to reach a particular goal that isn’t going to plan or is against our moral standards we might take a step back and focus on the individual tasks, details and process we are undertaking to achieve that goal.

Once the lower level tasks/details are modified so the goal again looks achievable we can then re-focus on the high level of meaning, the overall goal. Operating at either level has its draw-backs. At the higher level we may experience guilt from not adhering to our moral standards, i.e. looking out for our own interests at a negative cost to others. On the flip side operating at the lower level of meaning leaves us vulnerable to influence which may change how we act or what we think. We may change how we feel about a low level meaning which in turn will change the high level meaning once we return to focus at that level. There can also be differences of meaning within the high and low levels of meaning with important consequences. Dweck’s (1996) research has shown there are two styles we use:
 * Entity theorists - believes that traits are fixed and stable. They don’t like criticism and or negative feedback at all as they think any bad traits are permanent. They stick to things they are good at and easy tasks so they are guaranteed success and therefore receive associated credit admiration. Failure is devastating to entity theorists and they often lead to learned helplessness; they just give up and don’t try thus avoiding failure and being labelled incompetent.
 * Incremental theorists – believe traits are subject to change and improvement. They enjoy learning and challenges and don’t mind criticism or failure as they believe they will improve. They enjoy difficult tasks as a means of learning. If an incremental theorists failure they just try harder.

Both entity theorists and incremental theorists apply their beliefs to others. They basically believe people have fixed and stable behaviour traits and should not be expected to change (internal attributions) or that people’s traits are constantly changing and are in a temporary state (external attributions).

Linking actions to a goal is an important type of meaning. A goal is an idea of some desired future state. The goal is the link between values and action and because we hold certain values they then need to be translated in to behaviour to uphold those values.

Inner processes as well as cultural factors influences our goals. Cultural factors often set out the variety of possible goals. For example in some countries it is not culturally acceptable for a woman to run for government. We choose from those set goals depending on our current personal circumstances, wants and needs. If we start working toward a goal and either fail to achieve it or discontinue pursuing the goal a Zeigarnick effect occurs. Our automatic system interrupts our conscious system bringing thought of the incomplete goal to our thoughts.

The goals we set need to be realistic to maintain motivation to achieve. If the goal is to simple interest may be lost whereas if the goal is to difficult it may be seen as unachievable and again interest is lost. Most of us have a hierarchy of goals which requires us to plan and undertake a series of short term smaller goals or tasks (proximal goals) in pursuit of the long term ultimate goal (distal goal). The planning process of setting the proximal goals increase the chances of success in reaching the distal goal.

There are however, downsides to making plans. If the plans are inflexible they may be discouraging as they leave no scope for change. Also, making those plans can be very tiresome and time consuming which can also be discouraging. Another problem with making plans is they can be overly optimistic not allowing for unexpected issues i.e. setting deadlines that aren’t achievable because materials or resources aren’t available. This optimism is called the planning fallacy. We tend to be overly optimistic when looking long term because we look at the desirability whereas short term decisions tend to more realistic because we focus on the feasibility.

Self-determination theory
Freedom of action; people need feel they have some degree of autonomy and internal motivation. This means some of our motivation must come from our inner drives and not external forces. This autonomy leads to greater self-esteem and overall well-being from the knowledge it is an achievement of our own. Reaching goals as a result through extrinsic motivation is far less rewarding.

Interestingly we perform better in a stressful situation if we believe we have some control over the situation even if we don’t. This is called the panic button effect.

Making Choices
There are two main steps when making a choice if we are presented with a great range of options. A fairly quick short-list is drawn from the huge array of choices which excludes many of the options leaving only a select few. The select few are looked at in detail until a final choice is made. Some of the major patterns guiding choice are: Temporal discounting – immediate reward over future gain even if the future gain out weights the immediate reward.
 * Risk aversion – making choices to reduce risk by focusing on the possible losses more than the possible gain.
 * The certainty effect – giving greater weight, we often give undue greater weight to certain outcomes than to possibilities even if those possibilities are highly unlikely.
 * Keeping options open – postponing difficult decisions which can lead to decision avoidance if that final decision isn’t made. Decision avoidance is also known as the status quo bias which is the decision to keep things the way they were without change.

Haselton & Buss’ (2000) error management theory suggests both men and women minimise costly errors but because men tend to focus more on temporal discounting when making choices the errors differ. Another pattern that leads to no change is the omission bias or default option which is making the choice that requires no action at all and therefore avoids regret if a wrong decision is made.

Jack Brehm’s (1966) reactance theory is very similar to reverse psychology in as much as it we need to have freedom of choice and have a adverse negative reaction if choices are taken away by others or external forces. This reactance results in three main reactions:
 * desire for the choice no longer available increases
 * reasserting your freedom by trying to reclaim it
 * an aggressive reaction toward the person or cause of your restricted freedom


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User:Sandy~enwikiversity/Week 3 Social Thinking

Week 3 - Social Thinking
Lecture 3 Notes

Social Thinking Overview

 * How we intake and store information about our society.
 * Why people behave the way they do.
 * We combine what we perceive through our senses with our cognitive stores and information biases.
 * Communication - Exploring the different components of verbal and non-verbal communication
 * Impressions – formation and management, cognitive map

Duplex mind automatic fast process going on (largely unconscious) cognitive miser – use as little cognitive effort as possible Knowledge Structures – pre-programmed through environmental/social learning

Schemas
 * how we categorise things
 * descriptors of events and roles people play (mostly unconscious)
 * can be objects, concepts and relationships with other objects and concepts
 * schemata (plural)
 * no effort thought
 * exceptions to the rule can interfere with and prevents the uptake of new information (proactive interference)
 * stuck in an overreliance on automatic information

Scripts
 * social situations
 * the way of interacting
 * comes from exposure, experience and social learning through TV examples, watching other people
 * pre-programmed to acquire the cultural script

Stereotyping
 * takes schemas a bit further
 * cognition efficient thing to do
 * the normative characteristics ignoring the more general characteristics
 * exaggerated
 * calling on pre-coded procedures

Social Perception
Any social objects perception can be altered through interpretation. A social object doesn’t really exist in a meaningful way without context.
 * Priming – once learnt it is more readily accessible
 * Framing – social context

Typical Vs Unusual Events
 * Consistent events tend to rely on our schemas, scripts and stereotyping.
 * Unusual events require cognitive effort to explain causes of behaviour to regain a sense of control and predictability.
 * We think we can understand why people behave the way they do.
 * Why do we make these attributions (cause)?
 * For a sense of cognitive control
 * We don’t need to use to much energy trying to understand what’s going on

Attribution Theory
A sense of control over our social environment
 * How the perceiver uses information for explanation of events and causes
 * We perceive behaviour as being caused, even to inanimate objects
 * Both disposition and situation can cause behaviour
 * Personality motivation of the actor
 * Correspondent Influence Theory - When we believe a behaviour corresponds with a person’s internal belief.
 * Correspondent Bias – Underlying tendency to overestimate internal factors and underestimate cause being external.

Fundamental Attribution Bias Behaviour is more noticeable than situational factors. We are lazy and don’t want to think about the situational factors. If we aren’t cognitively busy we are likely to look at the situational information. If we are busy we tend to go for fundamental attribution error (FAE) There are cross cultural variations in attribution bias

'Actor/Observer bias
 * Actor – attribute behaviour to situations
 * Observer – attribution behaviour to disposition (person/individual)

Ultimate Attribution Error
 * FAE – individual to individual also to in-groups and out-groups
 * Bias towards our groups and against out-groups
 * Attitudes Beliefs and Consistency

Causes of behaviour
 * inside - internal, the person
 * external - others, the situation, the environment

Attitudes, Beliefs & Consistency
Attitudes Like or dislike to an object or issues

Beliefs
 * Sets of information we hold about objects or issues.
 * This information may be true or untrue
 * Why? It helps us to deal with the complexities of the world and it helps us to make quick decisions

Implicit Attitude Automatic evaluation respose

Explicit Attitude Our conscious evaluation response

Some attitudes are not visible, especially if they are socially unacceptable.

Mere Exposure Effect
If seen once we are more likely to become familiar with it and therefore more likely to have a positive attitude e.g. exposure to a product or brand may influence our decision on which brand to purchase.. An exception to this is if we have an initial negative reaction repeated exposure is unlikely to change this view.

Classical Conditioning Attitudes can be attributed to conditioning stimulous i.e. celebrities linked to products

Operant Conditioning Reward or punishment for a certain behaviour.

Social Learning
We don't have to learn something ourselves to through experience. We can learn through observing others e.g. exposure to a conversation.

Polarisation
An extension of exposure effect. Over time we are more confident of our attitudes. They become more extreme whether they are positive or negative.

Consistency
Consistency between behaviour, cognition (feeling) and affect (belief) otherwise we experience cognitive dissonance. We like to maintain the same attitude and want others to see our attitude is consistent where we don't say one thing and do another. We also like consistent self-presentation. Our drive for consitency is automatic.

Errors in Attribution
Because it is difficult to understand the meaning behind behaviour we use shortcuts (heuristics). This leads to error and biases such as:

 * Fundamental Attribution Error - Behaviour due to personal disposition not situation
 * Actor-Observer Bias - Actor attributes behaviour to situation while observer attributes it to personal disposition
 * Self-Serving Bias - Personal sucess and other's failures are internal while personal failure and other's success are external. We take credit for our success and deny responsibility for failure
 * Ultimate Attribution Error - Internal atributions for ingroup success anf out group failure while in-group failure and out-group sucess is attributed to external causes

Attitudes and Beliefs
Attitudes
 * Evaluation towards some object which helps us deal with our complex world
 * Initial evaluations are immediate and unconscious

Dual Attitudes
 * Implicit attitude - automatic evaluated response (we may not be aware of all of our attitudes)
 * Explicit attitude - conscious evaluated response

Classical Conditioning - learning through repeated pairing of a neutral stimulus to give a conditioned resonse. Helps with the formation of implicit and explicit attitudes.

Operant Conditioning - learned behaviour through reward and punishment.

Polarisation - the more we think about an attitude the stronger it becomes

Social Influence and Persuasion
Efforts of individuals to change the beliefs, attitudes or behaviour of others

Social Influence There are two types:
 * Normative influence (public compliance) - inner beliefs that the group is wrong
 * Informal Influence (private acceptance) - inner belief that others are right

Persuasion Attempts to change a person's mind. There are three components:
 * Who - the source of the message - must be credible and likable to be believable
 * What - the actual message - miderate fear appeals and repetition are most persuasive
 * Whom - the audience - overheard messagaes are the most persuasive

The two routes of persuasion
 * Central rouet - conscious processing with careful consideration
 * Peripheral route - automatic processing influenced by a simple cue

Tutorial - Communication
The various levels of Communication

Shallow Greetings:
 * acknowledgement
 * small talk

Deeper Communication:
 * self disclosure
 * share informationand facts
 * thoughts emotions and feelings

Verbal Communication
 * words

Non-verbal communication
 * facial expressions (incl eye movement)
 * hands (gestures)
 * semiotic symbols
 * tone of voice

7% is verbal 93% is non-verbal 38% Tone of voice 55% Body language

Transmission Model of Communication Similar to computers communicating
 * encoding language → transmit frequency ↔ decoding language ← feedback understanding

There can be a breakdown at any of these levels:
 * the message may not be clear
 * there may be a large large amount of ‘noise’ (distraction) preventing the message frm getting through
 * the message may be decoded incorrectly and therefore the message misinterpreted
 * the message may be interpreted in a way that differs from its intention due to pre-existing schemas
 * feedback may not be giving leaving the understanding the message had been received as intended

Active listening - repeating back what you hear. A very important tool particularly for counsellors and dealing with children; acknowledgment being given that what has been said is understood.

../Week 4_Aggression/


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User talk:CsikosLo~enwikiversity

 Hello CsikosLo, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. After you leave a comment on a talk page, remember to sign and date; it helps everyone follow the threads of the discussion. The signature icon in the edit window makes it simple. To get started, you may

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Glad to see another European History person! There's lots of discussion going on at the School talk page that we'd love your opinion on (particularly relevant stuff is marked with an asterisk). The Jade Knight (d'viser) 10:02, 31 October 2008 (UTC)

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Hello,

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title: User talk:CsikosLo~enwikiversity


text:
User:WikimediaNotifier/notifications

}}

title: User:WikimediaNotifier/notifications


text:
Managing a private wiki/LocalSettings.php/Default user options

See http://www.mediawiki.org/wiki/Manual:$wgDefaultUserOptions for the manual.

Changing watchlist settings
A good way to help people stay involved is to set the default so that any page a user creates or edits gets added to their watchlist.

Add the following to LocalSettings:

DPL alternative
If DPL is installed (and results are not cached), whole categories an be watched (useful when using subpages).


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User:MandaG/Tutorial and lecture 4

Aggression

Defining aggression is an interesting and in my opinion impossible thing to do. I don't see how any explanation can cover the myriad of circumastances in what may be percieved to be aggression. Popular theories of aggression include: *An intentional behaviour, that causes harm and has a victim that want's to void harm. *Violence *Behaviour that results in the destruction of property or personal injury. While these explanations may seem plausible on the surface, when you add them together its like 2+2=5. For example two years ago I was participating in a mixed martial arts class and learnt the hard way how not to get out of a headlock. The result of this lesson was a broken neck and a lucky to be alive story. Now I did not interpret this act to be aggressive. The bloke who was my partner had no intention to cause me such personal injury even though the act was violent. So does this count as aggression or not? It meets one of the definitions but not another. Who decides what constitutes aggrassive behaviour and what does not? As aggression is considered to be antisocial behaviour and what is antisocial is decided by societal norms and regulations. So while aggression is taken to be a universal concept, what is considered to be aggresion is not.

Types of Aggression Hostile aggression *Hostile agression is characterised as being impulsive and oriented towards creating harm. I think in real world scenario's this type of aggression is best witnessed during or after a heated, emotional arguement. Instrumental agression *Intrumental aggression is a cold, pre-meditated act. Mob Hits, need I say more? Passive aggression *Passive aggression is the act of harming others by witholding a behaviour. Active Aggression *The act of harming others by performing an act. High school is like a ses pool for active aggression.

Inner Causes of Aggression
 * Frustration-Aggression Hypothesis: The text tells us about the frustration aggression theorywhich is basically stating that people become frustrated when something that blocks or interferes with a goal which always leads to aggression. This hypothesis is based on Freud's we seek pleasure and avoid pain. I think this is in part true however, the idea that frustration always leads to some form of aggression is not realistic. The frustration-aggression theory doesn't count for any choices we make. Sometimes to achieve one goal people need to forgoe another, or achieve something for the "greater good" we make sacrafices which sure are frustrating at times but does not lead to aggression (I guess that sort of thing is why we no longer use the words always or never).
 * Being in a bad mood: Baumeister and Bushman say that people who are angry, upset and distraught are more prone to aggressive behaviour. Well duh. More technically though it is due to the the idea that people who are in tose states of emotion may act out in the belief that it will make them feel better.

Interpersonal Causes of Aggression So What to Do To reduce aggression it is highly recommended that the individual is around positive role models in the hopes that perhaps modelling will occur for the better, or at least not give rise to aggressive behaviour. That to release feelings of aggression one finds a healthy outlet in which to do it ie: kick boxing Anger management.
 * Social Learning Theory: One of he most famous social modelling/learning theories is that of Mr. Albert Bandura and his BoBo doll experiment. In this experiment the viewing of aggressive behaviour towards the doll increased the chances of the observer repeating the said behaviour. This experiment was criticisede for its contrived conditions and on its external validity. However, I think we see examples of social modelling everyday, and if we see it other area's why not with aggression?
 * The text looks at displacement aggression or kicking the dog but isn't moving your aggression about something else not a cause?

title: User:MandaG/Tutorial and lecture 4


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User:Eas4200c.f08.blue.a/Lecture 26

Bidirectional Bending

$$M_{y}=\int_{aA}^{}{z\sigma _{xx}dA}$$

$$M_{z}=\int_{aA}^{}{y\sigma _{xx}dA}$$

Moment of Inertia Tensors:

$$I_{yy}=\int_{A}^{}{z^{2}dA}$$

$$I_{zz}=\int_{A}^{}{y^{2}dA}$$

$$I_{yz}=\int_{A}^{}{zydA}$$

Recall that, by Hooke's Law :

$$\sigma _{xx}=E\varepsilon _{xx}$$

With the above equations this becomes:

$$\sigma _{xx}=(\frac{I_{yy}M_{z}-I_{yz}M_{y}}{I_{yy}I_{zz}-(I_{yz})^{2}})y$$+$$(\frac{I_{zz}M_{y}-I_{yz}M_{z}}{I_{yy}I_{zz}-(I_{yz})^{2}})z$$

Where the denominator of both terms is the determinant of bottom right corner of the Moment of inertia matrix (shown below):

$$I=\begin{bmatrix} I_{11} &I_{22} &I_{13} \\ I_{21}&I_{22} &I_{23} \\ I_{31}& I_{32} & I_{33} \end{bmatrix}$$

In addition, the location of the neutral axis can be found by setting $$\sigma _{xx}=0$$.

$$\sigma _{xx}=(\frac{I_{yy}M_{z}-I_{yz}M_{y}}{I_{yy}I_{zz}-(I_{yz})^{2}})y$$+$$(\frac{I_{zz}M_{y}-I_{yz}M_{z}}{I_{yy}I_{zz}-(I_{yz})^{2}})z=M_{yy}+M_{zz}=0$$

Therefore, $$z=(-\frac{M_{y}}{M_{z}})y=\tan (\beta) y$$

title: User:Eas4200c.f08.blue.a/Lecture 26


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User:Eml4500.f08.FEABBQ.koby/HW5

=Rederivation of 3-D Matrix Relations= Axial F-D Relation

Now that a 3 dimensional system is being considered, there is now one more additional degree of freedom at each node. However, in any element there can still only be two forces and those forces always be in the direction of the element. This means that the relation of P to Q remains unchanged:

$$\begin{bmatrix} P^{(e)}_1\\ P^{e)}_2 \end{bmatrix} = k^{(e)}\begin{bmatrix} 1 & -1\\ -1 & 1 \end{bmatrix} \begin{bmatrix} q^{(e)}_1\\ q^{e)}_2 \end{bmatrix}$$

Element DOF's and Element Forces in global XYZ coordinates (6x1 matrices) $$ \mathbf{d^{(e)}}=\begin{bmatrix} d^{(e)}_1 \\ d^{(e)}_2 \\ d^{(e)}_3 \\ d^{(e)}_4 \\ d^{(e)}_5 \\ d^{(e)}_6 \end{bmatrix} $$

$$ \mathbf{f^{(e)}}=\begin{bmatrix} f^{(e)}_1 \\ f^{(e)}_2 \\ f^{(e)}_3 \\ f^{(e)}_4 \\ f^{(e)}_5 \\ f^{(e)}_6 \end{bmatrix} $$

=Notes=

$$\mathbf{\hat{W}}_{2x1}$$ = Virtual axial displacement, corresponding to $$\mathbf{q^{(e)}_{1x1}}$$

$$\mathbf{W}_{4x1}$$ = Virtual displacement in the global coordinate system.

Replace equations (3) and (4) into equation (2)

$$\left( \mathbf{T^{(e)}w} \right)\bullet \left[ \mathbf{\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-p^{(e)}} \right]=0$$ for all $$\mathbf{W}_{4x1}$$  (5)

Recall $$\mathbf{\left(AB \right)^{T}=B^TA^T}$$  (6)

Recall: $$\mathbf{a_{nx1} \cdot b_{nx1}=a^T_{1xn}b_{nx1}=S_{1x1}}$$ (7)

Apply (6) and (7) into equation (5):

$$\mathbf{\left( T^{(e)}w \right)^T} \left[ \mathbf{\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-p^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{ w^TT^{(e)T}} \left[ \mathbf{\hat{k}^{(e)}\left(T^{(e)}d^{(e)} \right)-p^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{ w^T}\cdot \left[ \mathbf{ \underbrace{ \left( T^{(e)T}\hat{k}^{(e)}T^{(e)} \right) }_{k^{(e)}} }d^{(e)} -\underbrace{ T^{(e)T}p^{(e)}}_{f^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{ w^T}\cdot \left[ \mathbf{{k}^{(e)}d^{(e)} -f^{(e)}} \right]=0_{1x1}$$

$$\Rightarrow \mathbf{{k}^{(e)}d^{(e)} =f^{(e)}}$$

The Continuous Case

So far we have only been analyzing discrete cases (with matrices). These cases are not continuous. The next phase of the class will look at the continuous case (with Partial Differential Equations, PDEs). There are many motivational problems such as an elastic bar with varying A(x), E(x), varying axial load (distributed and concentrating), and inertia forces (dynamics).

title: User:Eml4500.f08.FEABBQ.koby/HW5


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User:Eml4500.f08.ateam.boggs.t/week 9

Lecture 24

title: User:Eml4500.f08.ateam.boggs.t/week 9


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User:Eml4500.f08.ateam.boggs.t/Lecture 24

Proving Hookes Law through the Principle of Virtual Work
The equation $$ F = kd $$ can also be written as $$ kd - F = 0 $$ (Equation 3) or which is true for all values of W if $$ W(kd - F) = 0 $$ (Equation 4). This is a weak form of the Principle of Virtual Work. It is important to note that deriving Equation 4 from Equation 3 is a trivial solution, but deriving Equation 3 from Equation 4 is in fact Not trivial.

From the Force-Displacement relationship,

 * $$ F = kd $$

where F is a 6x1 matrix, k is a 6x6 matrix, and d is a 6x1 matrix, the Principle of Virtual Work can be derived as,

 * $$ W \cdot (kd - F) = 0 $$ (Equation 4)

This is true for all W. For this relationship to work W must be multiplied as a dot product, as 0 is not represented as a matrix. It is desired to show that Equation 3 can be derived from Equation 4 for all W.

To do this the first choice is to select W such that W1 = 1 and W2 through W6 = 0. W is originally defined as a 6x1 matrix but needs to be a 1x6 matrix so,

W^{T} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} $$

Equation 4 then becomes,

W \cdot (kd - F) = 1[\sum_{j=1}^6 k_{ij}d_{j} - F_{1}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{2}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{3}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{4}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{5}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{6}] = 0 $$

This becomes,

\sum_{j=1}^6 k_{ij}d_{j} - F_{1} = 0 $$

The second choice is to select 'W such that W1 equals 0, W2 equals 1, and W3 through W6 also equal 0. So,

W^{T} = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 & 0 \end{bmatrix} $$

Equation 4 then becomes,

W \cdot (kd - F) = 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{1}] + 1[\sum_{j=1}^6 k_{ij}d_{j} - F_{2}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{3}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{4}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{5}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{6}] = 0 $$

This becomes,

\sum_{j=1}^6 k_{ij}d_{j} - F_{2} = 0 $$

This repeats as follows.

Choice 3,

W^{T} = \begin{bmatrix} 0 & 0 & 1 & 0 & 0 & 0 \end{bmatrix} $$

Equation 4 then becomes,

W \cdot (kd - F) = 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{1}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{2}] + 1[\sum_{j=1}^6 k_{ij}d_{j} - F_{3}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{4}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{5}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{6}] = 0 $$

This becomes,

\sum_{j=1}^6 k_{ij}d_{j} - F_{3} = 0 $$

Choice 4,

W^{T} = \begin{bmatrix} 0 & 0 & 0 & 1 & 0 & 0 \end{bmatrix} $$

Equation 4 then becomes,

W \cdot (kd - F) = 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{1}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{2}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{3}] + 1[\sum_{j=1}^6 k_{ij}d_{j} - F_{4}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{5}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{6}] = 0 $$

This becomes,

\sum_{j=1}^6 k_{ij}d_{j} - F_{4} = 0 $$

Choice 5,

W^{T} = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 0 \end{bmatrix} $$

Equation 4 then becomes,

W \cdot (kd - F) = 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{1}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{2}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{3}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{4}] + 1[\sum_{j=1}^6 k_{ij}d_{j} - F_{5}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{6}] = 0 $$

This becomes,

\sum_{j=1}^6 k_{ij}d_{j} - F_{5} = 0 $$

Choice 6,

W^{T} = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} $$

Equation 4 then becomes,

W \cdot (kd - F) = 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{1}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{2}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{3}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{4}] + 0[\sum_{j=1}^6 k_{ij}d_{j} - F_{5}] + 1[\sum_{j=1}^6 k_{ij}d_{j} - F_{6}] = 0 $$

This becomes,

\sum_{j=1}^6 k_{ij}d_{j} - F_{6} = 0 $$

Hence $$ F = kd $$ (Equation 1).

To account for the boundary conditions of the two-bar truss where,

 * $$ d_{1} = d_{2} = d_{5} = d_{6} = 0 $$

the weighting coefficient must be "kinematically admissible" or cannot violate the boundary conditions. The weighting coefficients are the virtual displacements where,

 * $$ W_1 = W_2 = W_5 = W_6 = 0 $$

For all $$ W_3 $$ and $$ W_4 $$,

\begin{bmatrix} W_3 \\ W_4 \end{bmatrix} \cdot ( \begin{bmatrix}     k_{33} & k_{34} \\      k_{43} & k_{44} \end{bmatrix} \begin{bmatrix}      d_3 \\      d_4 \end{bmatrix} - \begin{bmatrix}      F_3 \\      F_4 \end{bmatrix} ) = 0 $$

Equation 1 can also be found through Equation 3 by adding F to both sides of the equation as shown below,

 * $$ kd - F = 0 $$

Adding F to both sides,

 * $$ kd - F + F = 0 + F $$

Here F cancels out on the left-hand side of the equation leaving,

 * $$ kd = F $$


text:
User:Eas4200c.f08.aero.e/Week 9

=Monday= Revisiting the road map,

Kinematic assumption (Compare to pg. 70)

4 - Stain components (Compare to pg. 70)

4 zero stress components (Compare to pg. 70)

$$ \epsilon - \sigma $$ relationship (2 forms: tensorial or engineering)

Rewrite $$ \epsilon - \sigma $$ relation

$$ \varepsilon_{ij} = \begin{bmatrix} {\underline A}_{3x3} & {0}_{3x3}\\ {0}_{3x3} & {\underline B}_{3x3}\\ \end{bmatrix} \sigma_{ij} $$

Let $$ \begin{bmatrix} {\underline A}_{3x3} & {0}_{3x3}\\ {0}_{3x3} & {\underline B}_{3x3}\\ \end{bmatrix} = \underline C $$

An advantage of using this form is that $$\underline C$$ can be easily inverted.

$$ \sigma_{ij} = \begin{bmatrix} {\underline A}^{-1} & \underline 0\\ \underline 0 & {\underline B}^{-1}\\ \end{bmatrix} \varepsilon_{ij} $$

$$ \begin{bmatrix} {\underline A}^{-1} & \underline 0\\ \underline 0 & {\underline B}^{-1}\\ \end{bmatrix} = \underline {C}^{-1} $$

When the two matrices are dotted into each other we get the identity matrix.

$$\underline C \underline {C}^{-1} = \underline I$$

Goal: prove 0 stress components.

$$\sigma _{xx}, \sigma _{yy} , \sigma _{zz}, \tau_{yz} = 0$$

$$\epsilon _{xx}, \varepsilon_{yy} , \varepsilon_{zz}, \gamma_{yz} = 0$$

This implies,

$$ \sigma_{ij} = \begin{bmatrix} {\underline A}^{-1} & \underline 0\\ \underline 0 & {\underline B}^{-1}\\ \end{bmatrix} \begin{bmatrix} 0\\ 0\\ 0\\ 0\\ \varepsilon_{31}\\ \varepsilon_{12}\\ \end{bmatrix} $$

(Note, using numerical notation as introduced in MIT OCW assigned reading)

$$\sigma _{11}, \sigma _{22} , \sigma _{33} = 0$$

$$\sigma _{23} = 2G\varepsilon_{23} = \frac{1}{2}\gamma_{23} = \frac{1}{2}\gamma_{yz}$$

$$\sigma _{31} = 2G\varepsilon_{31}$$

Roadmap step C:

Implicit reading: Section 2.4 in book (Non-Uniform stress field)

Consider 1-D case as a model:

=Wednesday=

 Exam 2

=Friday=

 Homecoming

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User:Eas4200c.f08.aero.e/Week 10

=Monday=

Birectional Bending: See sections 4.2 & 4.1 and Example 4.1 on Page 124 of the text book.

Homework 5. Stringers @ points B,E,H,F on lecture 22-1

*Skin and span web do not contribute in beding; only stringers

*Need areas of stringers $$A_{B}, A_{E}, A_{H}, A_{F}$$

Bidirectional bending recipe: (no symmetry)

$$M_{y}=\int_{A}^{}{}Z\sigma _{xx}dA$$ (sec. 4.26)

Moment of inertia tensor: $$I_{y}=I_{yy} I_{x}=I_{xx}  I_{z}=I_{zz}$$

$$I_{y}=\int_{A}^{}{}z^2dA$$ (sec. 4.28a)

$$I_{Z}$$ (sec. 4.28b)

$$I_{yz}=\int_{A}^{}{}yzdA$$ (sec.4.28c)

$$\sigma_{xx}=Eepsilon_{xx}=$$

=Wednesday= HW.5 continued from pg (24-3)

Equation of Equilibrium (Stresses)

Goal: $$\frac{\delta \sigma_{yx}}{\delta y} + \frac{\delta \sigma_{zx}}{\delta z}=0$$ (compare with 3.14 in book)

This equation derived in a section detailing the governing equations of applying torque to uniform bars. Essentially, cross-sections of a uniform body undergoing an angular displacement can be treated as rigid bodies to which a rotation has been applied. Because the properties associated with each cross-section are independent of z, the strains and all but the above shear stress values equal zero.

Rewriting the equilibrium equation in indicial notation:

$$\frac{\delta \sigma_{21}}{\delta x_2} + \frac{\delta \sigma_{31}}{\delta x_3}=0$$

Recall: (Sec 2.4)

$$\frac{\delta \sigma_{zx}}{\delta x} + \frac{\delta \sigma_{yz}}{\delta y} + \frac{\delta \sigma_{zx}}{\delta z}=0$$  (Eq. 2-21)

Similarly, see (2.22 and 2.23)

In indicial notation, (2.21, 2.22, and 2.23):

$$\sum_{i=1}^{3}{\frac{\delta \sigma_{ij}}{\delta x_i}=0 }$$ for j=1,2,3

=Friday=

Derivation of equation:

$$\sum_{i=1}^{3}{}\frac{\partial \sigma _{ij}}{\partial x_{i}}=0 $$ for j = 1,2,3

We will do the 1-D case first for simplification.

And we will be referring to the following figure:

$$\sum{F_{x}} = 0 = -\sigma (x)A + \sigma (x + dx)A + F(x)dx$$

$$0 = A\left[ \sigma (x+dx)-\sigma (x)+F(x)dx\right] $$

Now, if we use Taylor series expansion the above expression can be converted to another useful expression as follows:

$$ 0 = \left[ \sigma (x + dx) - \sigma (x)\right]A + F(x)dx $$ = $$\frac{\partial  \sigma (x)dx}{\partial x} + H.O.T = 0$$

Where H.O.T = Higher Order Terms

Recall: $$f(x+dx) = f(x) + \frac{df(x)dx}{dx} + \frac{1d^{2}f(x)dx}{2dx^{2}} + ...$$ and we neglect H.O.T

Then, we get:

$$\frac{d\sigma }{dx} + \frac{f(x)}{A} = 0$$ = Applied load

Now, non-uniform stress field in 3-D, but without applied load and focusing on the x-direction only (i.e: without the other stress components to avoid cluttering of the figure):

Where we will use the following convention: σij, where i indicates that is normal to the facet and j indicates the direction of stress.

Then we obtain:

$$\sum{}F_{x} = 0 = dy dz[-\sigma _{xx}(x,y,z)+\sigma _{xx}(x+dx,y,z)]+ dzdx[-\sigma _{yz}(x,y,z)+\sigma _{yx}(x,y+dy,z)]+ dxdy[-\sigma _{zx}(x,y,z)+\sigma _{zx}(x,y,z+dz)]$$

$$(dx)(dy)(dz)[\frac{\partial \sigma _{xx}}{\partial x}+\frac{\partial \sigma _{yx}}{\partial y}+\frac{\partial \sigma _{xz}}{\partial z}]= 0$$

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