In this chapter, we describe how to obtain elementary stiffness and mass matrices for the right beam element, according to the Euler (POU_D_E) or Timoshenko (POU_D_T) model. Stiffness matrices are calculated with option “RIGI_MECA”, and mass matrices with option “MASS_MECA” for the coherent matrix, and option “MASS_MECA_DIAG” for the diagonalized mass matrix.
4.1. Longitudinal traction movement - compression
A difficulty in writing the variational formulation comes from the fact that structures composed of beams may have concentrated loads (similar to Dirac). Balance equation 3.1.1-1 should be replaced by:
\[\ frac {dN} {\ mathrm {xx}} (x) + {f}} _ {\ mathrm {ext}} (x) +\ sum _ {i=1} ^ {N} {f} _ {f} _ {f} _ {i} _ {i} {f} _ {f} _ {i} (x) =0\]
For simplicity, inertial forces were omitted, which would be treated in the same way as external forces \({f}_{\mathrm{ext}}\).
\({\delta }_{i}\) represents the Dirac function located at point \(i\), the \({f}_{i}^{c}\) are the concentrated forces applied to the beam.
For the application of the finite element method, the equilibrium equation must be written in the form of the principle of virtual works, which is in this case:
\[\]
: label: 4.1-1
underset {G} {int} Nfrac {dv} {frac {dv}} {text {xx}} =underset {G} {int} {f} _ {text {ext}}}cdot vfrac {dv}} {dv} {dv} {text {dv}}}cdot vtext {dv} {dv} {text {ext}}}}cdot vtext {dv} {text {dv}}}cdot vtext {dv} {text {dv}} {dv} {text {ext}}}}cdot vfrac {dv} {dv} {text {dv}} {dv} {dv)
Any confusion being ruled out, \({\delta }_{i}\) designates the Dirac measure associated with the point \(i\), \(y\) is any kinematically admissible longitudinal field of displacement.
In practice, it is assumed that there is no concentrated force inside the beam elements, but only at the end nodes.
4.1.1. Determination of the stiffness matrix
It corresponds to the expression of the virtual work of internal forces according to a given displacement. That is to say:
\({\int }_{0}^{L}N\frac{dv}{\mathrm{dx}}\mathrm{dx}\) for an element of length \(L\).
We introduce the elastic behavior relationship: \(N(x)=\text{ES}\frac{\text{d}u}{\text{dx}}\)
By choosing for test functions: \(v(x)={\xi }_{1}(x)=1-\frac{x}{L}\) and \(v(x)={\xi }_{2}(x)=\frac{x}{L}\)
corresponding respectively to the degrees of freedom \({u}_{1}\) and \({u}_{2}\) of the two nodes of the element,
we obtain directly:
\[{\ int} _ {0} ^ {L} N\ frac {d {\ xi} _ {1}} {\ text {xx}}\ mathrm {xx} = {\ int} _ {0} ^ {L} -\ frac {\ text {S}} {L}} {L}\ frac {\ text {of}}}\ mathrm {xx}}\ mathrm {xx}} {L} -\ frac {\ text {ex}} =-\ frac {\ text {ES}} {L}\ left [u (L) -u (0)\ right]\]
and
\[{\ int} _ {0} ^ {L} N\ frac {d {\ xi} _ {2}} {\ text {xx}}\ mathrm {xx} = {\ int} _ {0} ^ {L}\ frac {\ text {S}}\ frac {\ text {ES}} {L}}\ frac {\ text {of}} {\ text {xx}} =\ frac {\ text {ES}} {L}\ left [u (L) -u (0)\ right]\]
The stiffness matrix of the element is therefore:
\[\begin{split}K=\ frac {\ text {ES}} {L}} {L}\ left [\ begin {array} {cc} 1& -1\\ -1& 1\ end {array}\ right]\end{split}\]
Note
In the expression of the virtual work of inner efforts, \(u\) only intervenes for \(u(0)\) and \(u(L)\)
\(u\) has not been discretized within the element. This is why the element is called « exact »: the exact solution is obtained at the nodes, but only at the nodes.
4.1.2. Determination of the second member
The second member is the expression of the virtual work of applied efforts. The second member associated with distributed loading and the test functions previously introduced is:
\[\begin{split}\ left [\ begin {array} {c} {f} _ {1} _ {1}\\ {f} _ {2}\ end {array}\ right]\ text {with} {f} _ {1} = {\ int} _ {0} _ {0} _ {0}} ^ {1} {1} {f} _ {1} {f} _ {f} _ {2}\ end {array}\ right]\ text {with} {f} _ {1} = {\ int} _ {1} = {\ int} _ {0} _ {0}} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {1} ^ {)\ mathit {xx}\ text {and} {f} _ {2} _ {2} = {\ int} _ {0} ^ {1} {f} _ {\ text {ext}}}\ left (x\ right)\ left (x\ right)\ frac {x} _ {2} = {\ int} _ {0} ^ {1} {f} _ {\ text {ext}}}\ left (x\ right)\ frac {x} _ {x}\end{split}\]
Note
In AFFE_CHAR_MECA_F, :math:`{f}_{mathit{ext}}` can be introduced as any function in :math:`x`. In terms of the calculation of :math:`{f}_{1}` and :math:`{f}_{2}`, on the other hand, the integration is done by assuming that :math:`{f}_{mathrm{ext}}` varies linearly between the values taken at the end nodes. If we have to model a non-linear distributed load variation, we must then discretize more finely.
But let’s emphasize the fact that regardless of the form of :math:`{f}_{mathrm{ext}}(x)` (polynomial or other), if we know how to calculate the integrals :math:`{f}_{1}` and :math:`{f}_{2}` exactly, the solution of the static problem will be exact at the crux of the problem.
The virtual work of concentrated forces (given by hypothesis at the nodes of the elements) does not intervene directly at the level of the element.
These concentrated forces are introduced in the form of nodal forces, directly into the assembled vector of the second limb.
4.1.3. Calculation of the nodal forces at the nodes of the beam
The complete Principle of Virtual Works is in fact written on the assembled system. On the other hand, by writing the integration formula by part over the whole structure (beam \(\left[{x}_{0},{x}_{1}\right]\)):
\[\]
: label: 4.1.3-1
{int} _ {Gamma} N {v} _ {, x}text {xx} =left [Nleft ({x} _ {1}right) vleft ({x} _ {1}right) -Nleft ({1}right) -Nleft ({x}}right) -Nleft ({x}}right)right) -Nleft ({x}\ right) -Nleft ({1}right) -Nleft ({x}\ right) -Nleft ({1}right) -Nleft ({x}\ right)right) -Nleft ({x}\ right) -Nleft ({x}\ right)right) -Nleft ({x}} ^ {N} {left [mid Nmidright]} _ {i} {delta} _ {i}left (vright) -sum _ {j=1} ^ {M}underset {{Gamma}underset {{Gamma}} _ {underset {Gamma}} _ {M}underset {Gamma} _ {M}underset {Gamma} _ {M}underset {Gamma} _ {Gamma} _ {M}underset {Gamma} _ {Gamma} _ {M}underset {Gamma} _ {Gamma} _ {x} vunderset {{Gamma}} _ {
\({\Gamma }_{j}\) representing all the intervals without a discontinuity of normal effort, therefore without concentrated force, and \({\left[\mid N\mid \right]}_{i}\) the jumps of \(N\) between these intervals.
Indeed, by comparing this expression to the Principle of Virtual Works, we find, for each concentrated load (by choosing the appropriate test functions \(v\)):
\[i=\ mathrm {1,} N {\ left [\ mid N\ mid\ right]} _ {i} = {f} _ {i} _ {i}} ^ {c}\]
Each finite element of a beam is hypothetically an interval without discontinuity. There can therefore be a discontinuity of internal forces \(N\) from one element to another if there is a force concentrated on the node connecting the two elements.
The internal forces for an element are determined as follows. The equilibrium equation within an element is:
\[{N} _ {, x} + {f} ^ {\ text {rep}} =0\]
The formula for integration by parts 4.1.3-1 on the element gives:
\[{\ int} _ {0} ^ {L} N (x) {v} _ {v} _ {v} _ {, x} =\ left [N (L) v (L) -N (0) v (0)\ right] + {\ int}\ right] + {\ int}\ right] + {\ int} _ {\ int} _ {0} ^ {0} ^ {L} {f} _ {\ text {ext}}} (x) v (x)\ right] + {\ int} _ {0} _ {0} ^ {0} ^ {L} {f} _ {\ text {ext}} (x) v (x)\ right] + {\ int} _ {0} _ {0} ^ {0} ^ {L} {f} _\]
By considering \(N(L)\) and \(N(0)\) as data, we could have obtained this formula directly from the Principle of Virtual Works 4.1-1.
By still using the test functions:
\[v (x) = {\ xi} _ {1} (x) =1-\ frac {x} {L}\ text {and} v (x) = {\ xi} _ {2} (x) =\ frac {x} {L}\]
We get:
\[\begin{split}\ begin {array} {}
-\ frac {\ text {ES}} {L} {L}\ left [u (L) -u (0)\ right] =-N (0) + {f} _ {1}\\
\ text {}\ frac {\ text {ES}}} {L} {L}\ left [u (L) -u (0)\ right] =N (L) + {f} _ {2}\\
\ text {be}\ left [\ begin {array} {c}
-N (0)\\
N (L)
\ end {array}\ right] =\ left [K\ right]\ left [\ begin {array} {c}
u (0)\\
U (L)
\ end {array}\ right] -\ left [\ left [\ begin {array} {c}
{f} _ {1}\\
{f} _ {2}
\ end {array}\ right]\ end {array}\end{split}\]
That is to say, the internal forces are obtained by subtracting from the product \(K\cdot U\) the nodal forces equivalent to the distributed loads \({f}_{\mathrm{ext}}\).
It is also observed that they are of the opposite sign. For the sign to be the same from one element to another, you must therefore change the sign of \(N(0)\) calculated by this method. This is what is done by calculating option EFGE_ELNO.
4.1.4. Determination of the mass matrix
To be consistent with the stiffness matrix, the mass matrix is determined from the same test functions. However, it is no longer possible to calculate the associated nodal forces exactly without making an assumption about the shape of the solution. The calculation of the mass matrix will cause a discretization error.
A dynamic calculation will therefore require a discretization of the beam structure into small elements, which is not the case for a static calculation. It goes without saying that in the case of a dynamic calculation, the calculation of the forces that will be carried out as in [§ 4.1.2] by subtracting the nodal forces of inertia is also approximate. Solution \(u\) is chosen in the space generated by the test functions (i.e. polynomials with degrees at most equal to 1):
\[u=u (0) {\ xi} _ {1} (x) +u (L) {\ xi} _ {2} (x)\]
The mass matrix appears in the expression of virtual work due to inertial forces:
\[\begin{split}W= {V} ^ {T} M\ ddot {U}, U= (\ begin {array} {c} {u} _ {1}\\ {u} _ {2}\ end {array})\end{split}\]
The work is also written as:
\[W= {\ int} _ {0} ^ {L} ^ {L} V (x) {\ rho} _ {m}\ ddot {u} (x, t)\ mathrm {dx}\]
with \({\rho }_{m}={\int }_{s}\rho \mathrm{dS}=\rho S\) in the case of homogeneous material.
Taking \(u(x,t)={\xi }_{1}(x){u}_{1}(t)+{\xi }_{2}(x){u}_{2}(t)\), we have:
\[\begin{split}W= {\ int} _ {0} ^ {L} {L} {V} {V} ^ {V} ^ {T} ^ {T}\ rho\ text {S} (c} {\ xi} _ {1} (x)\\ {\ xi} (x) _ {2} (xi) _ {2} (xi) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x))\ ddot {U}\ text {xx}\end{split}\]
\[\begin{split}\ text {W =} {V} ^ {T} (\ rho S {\ int}} _ {0} ^ {L} (\ begin {array} {\ xi} _ {1} (x)\\ {\ xi} _ {\ xi} _ {2}} _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x) _ {2} (x))\ ma* thrm {xx})\ ddot {U}\end{split}\]
So the mass matrix is written as:
\[\begin{split}M=\ rho\ text {S} (\ begin {array} {cc} {cc} {\ int} _ {0} ^ {L} {\ xi} _ {1} ^ {2}\ text {xx} & {\ int} & {\ int} _ {0}} _ {0} ^ {L} {L} {\ xi} _ {1} {xi} _ {2}\ text {xx}\\ {\ int} & {\ int} & {\ int} _ {0} ^ {L} _ {0} ^ {L} _ {0} ^ {L} _ {0} ^ {L}} {\ xi} _ {1} {\ xi} _ {2}\ text {xx} & {\ int} _ {0} ^ {L} {\ xi} _ {2} ^ {2} _ {2} ^ {2}\ text {xx}\ text {xx}\ end {array})\end{split}\]
and calculations made:
\[\begin{split}\ mathrm {M} =\ frac {\ rho\ text {SL}}} {6}\ left (\ begin {array} {cc} 2& 1\\ 1& 2\ end {array}\ right)\end{split}\]
corresponding respectively to the degrees of freedom \({u}_{1}\) and \({u}_{2}\) of the two nodes of the element.
4.2. Free torsional movement around the longitudinal axis
The problem is similar to that of traction compression. For a \(\Gamma\) beam, loaded with distributed torsional moments \({\Gamma }_{x}(x)\) and concentrated moments \({\Gamma }_{i}^{c}\), the principle of virtual work is written:
\[{\ int} _ {\ Gamma} {M} {M} _ {x}\ frac {d\ Psi}\ frac {d\ Psi}} {\ text {xx} = {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {x} _ {x} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} _ {\ Gamma} delta} _ {i}\ left (\ Psi\ right),\ forall\ Psi\]
The law of behavior is:
\[{M} _ {x}\ left (x\ right) =G\ cdot C\ cdot\ frac {d {\ theta} _ {x}} {\ text {xx}} {\ text {xx}}\]
With the exception of variables, this equation has the same form as that of the pull-compression movement. Using the same reasoning, we obtain the same expressions for the elementary mass and stiffness matrices, namely:
\[\begin{split}\ mathrm {K} =\ frac {\ text {G C}}} {L}} {L}\ left (\ begin {array} {cc} 1& -1\\ -1& 1\ end {array}\ right)\ mathit {and}\ mathrm {M}} =\ frac {\ rho} {M} =\ frac {\ rho} {M} =\ frac {\ rho\ text {c L}} {cc} {6}\ left (\ begin {array}\\ right)\ mathit {and}}\ mathrm {M}} =\ frac {\ rho} {M} =\ frac {\ rho} {M} =\ frac {\ rho\ text {C L}} {6}\ left (\ begin {array}\\ 1& 2\ end {array}\ right)\end{split}\]
corresponding respectively to the degrees of freedom \({\theta }_{{x}_{1}}\) and \({\theta }_{{x}_{2}}\) of the two nodes of the element.
As previously, the calculation of the mass matrix required discretizing the solution field.
The second limb, due to the distributed \({\Gamma }_{x}\) torque, is obtained in the same way as for the traction-compression movement:
\[\begin{split}\ left (\ begin {array} {c} {\ int} _ {\ int} _ {0}} ^ {L} {\ xi} _ {1}\ left (x\ right) {\ Gamma} _ {x}\ text {dx}\\ xx}\\ {int}\\ int}\\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {0} ^ {0} ^ {L} {L} {\ xi} _ {2}\ left (x\ right) {\ Gamma} _ {x}\ text {xx}\ end {array}\ right)\ text {}\ begin {array} {c} {\ xi} {\ xi} _ {1}\ left (x\ right) =1-\ frac {x} {L}\\ {\ xi} _ {2} _ {2}\ left (x\ right) =\ frac {x} {L}\ end {array} _ {2}\ left (x\ right) =\ frac {x} {L} {L}\ end {array} _ {2}\ left (x\ right)\end{split}\]
4.3. Flexion movement
Here we are placing ourselves in the framework of a straight beam with a constant cross section of the Timoshenko type. We take into account the effects of transverse shear. The Euler-Bernoulli beam will then be treated by simplifying the Timoshenko equations.
The description of flexure is more complex than the previous movements, but a careful choice of test functions will allow us to obtain results of the same shape.
4.3.1. In-plane bending (xOz)
With obvious notations and by not initially focusing, as in the previous cases, on inertial forces, the principle of virtual work is written for the flexure movement in plane \((\mathrm{x0z})\):
\[\]
: label: 4.3.1-1
{int} _ {Gamma} {V} _ {z} _ {z} (Psi”+omega) + {M} _ {y}omega “= {int} _ {Gamma} ({t} _ {{z}} _ {{z} _ {{z} _ {text {z}} _ {{z}} _ {{z} _ {{z}} _ {{z} _ {{z} _ {{z} _ {z} _ {{z} _ {{z} _ {z} _ {{z} _ {{z} _ {z} _ {{z} _ {{z} _ {z} _ {{z} _ {{z} _ {z} _ {{z} _ {z} _ {{z} _ {{z} ^ {N} {t} _ {i} ^ {c} {c} {delta} _ {i} (Psi) + {m} _ {i} ^ {c} {delta}} {delta} _ {c} _ {i} _ {i} (omega)
for any cinematically eligible \((\Psi ,\omega )\).
The stiffness matrix is deduced from the expression of the virtual work of internal forces that we will explain by using the behavioral relationship and then by integrating by parts:
\[\begin{split}{\ int} _ {\ Gamma} {V} {V} _ {z} (\ Psi'+\ omega) + {M} _ {y}\ omeg'& = {\ int} _ {k} _ {k} _ {z} _ {z} SG (w'+ {z}} SG (w'+\ theta} _ {y}) (w'+\ theta} _ {y}) (\ PSI '+\ omega) +E {I} _ {y} _ {\ theta} _ {y} ^ {'}\ omega'\\
& = {k} _ {z} SG\ left [w (L) [w (L) (W) (L) +\ omega (L)) -w (0) (\ Psi '(0) +\ omega (0))\ right]
- {\ int} _ {\ Gamma} {k} {k} _ {z} SGw (\ PSI "+\ omega ') + {\ int} _ {\ Gamma} {k} _ {z} SG {\ theta} _ {z} SG {\ theta} _ {z} _ {z} _ {z} SG {\ theta} _ {y} _ {y} (\ PSI '+\ omega)\
& + {EI} _ {y}\ left [{\ theta}\ left [{\ theta}} _ {\ theta} _ {y} (0)\ omega '(0)\ right] - {\ int}] - {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int} _ {\ int}\end{split}\]
The test functions that we will choose will « verify the equilibrium equations without a second member », that is to say 3.3.1-2 and 3.3.1-3:
\[\]
: label: 4.3.1-2
- begin {cases}
Psi « +omega “=0\
EI_ {y}omega « - {k} _ {z}mathrm {SG} (PSI “+omega) =0
end {cases}
Under these conditions, the nodal forces, an expression of the work of the internal forces in these given virtual displacements, are expressed exactly, without hypotheses on the form of the solution, as a function of the displacements at the end of the beam as in the previous cases:
\[\]
: label: 4.3.1-3
underset {Gamma} {int} {V} {V} _ {z} (Psi”+omega) + {M} _ {y}omeg”= {k} _ {z}mathrm {SG}left [w (L)}left [w (L)}left [w (L)} _ {g}}left [w (L)] (L) _ {z}left [w (L)}left [w (L)] (L) (Psi “(L) +omega (L)) -w (0) (0) +omega (0) +omega (0))right] + {mathrm {EI}} _ {y}left [{theta} _ {theta} _ {y} (L)omega” (L) - {theta} _ {y} (0) (0)omega “(0)right]
Note
It is clear that condition 4.3.1-2 leads to test functions that depend explicitly on the geometric and material characteristics of the beam, but this is nothing annoying.
The pairs of test functions chosen are: \((\Psi ,\omega )=({\xi }_{i}\text{,}{\xi }_{i+4})i=\left[\mathrm{1,}4\right]\)
where, having noted \({\phi }_{y}=\frac{12{EI}_{y}}{{k}_{z}{\text{SGL}}^{2}}\), the functions \({\xi }_{i}\) are defined by:
Movations:
\[\]
: label: 4.3.1-4
{xi} _ {1} (x) & =frac {1} {1} {1+ {varphi}} _ {y}}left [2 {left (frac {x} {L}right)} ^ {3}right)} ^ {x}right)} ^ {2} - {varphi} {L} {L}right)} ^ {2} - {varphi} _ {y}right)} ^ {y}frac {x} {L} +left (1+ {varphi} _ {y}right)right]\
{xi} _ {2} (x) & =frac {L} {1+ {varphi} _ {y}}left [- {left (frac {x} {L}right)} ^ {3}right)} ^ {3} +frac {L}} {4+ {varphi} {4+ {varphi} _ {y}}} {2} {left (frac {x} {L}right)} ^ {L}right)} ^ {2} -frac {2+ {varphi} _ {y}}} {2}left (frac {x} {L}right)right)right]\
{xi} _ {3} (x) & =frac {1} {1} {1+ {varphi} _ {y}}text {}left [-2 {left (frac {x} {L} {L} {L}right)} ^ {x} {L}right)} ^ {2} + {varphi} {L} {L} {L} {L}right)} ^ {2} + {varphi} {L} {L}right)} ^ {2} + {varphi} {L} {L}right)} ^ {2} + {varphi} {L} {L}right)} ^ {2} + {varphi} {L} {L} {L}right)} ^ {2} + (frac {x} {L}right)right]\
{xi} _ {4} (x) & =frac {L} {1+ {varphi} _ {y}}left [- {left (frac {x} {L}right)} ^ {3}right)} ^ {3} +frac {3} +frac {L}} +frac {L} {L}right)} +frac {2- {varphi} _ {y}}} {2} {left (frac {x} {L}right)} ^ {2} +frac {{varphi} _ {y}} {y}} {2}left (frac {x} {L} {L}right)right)right]
Rotations:
\[\]
: label: 4.3.1-5
{xi} _ {5} (x) & =frac {6} {Lleft (1+ {varphi} _ {y}right)}frac {x} {L} {L}left [1-frac {x} {L}right]\
{xi} _ {6} (x) & =frac {1} {1} {1+ {varphi} _ {y}}text {}left [3 {left (frac {x} {L} {L}right)} {L}right)} =right)} =right)left (frac {x} {L}right)}right)left (1+ {varphi} _ {y}right)right]\
{xi} _ {7} (x) & =frac {-6} {Lleft (1+ {varphi} _ {y}right)}frac {x} {L} {L}left [1-frac {x} {L}right]\
{xi} _ {8} (x) & =frac {1} {1} {1+ {varphi} _ {y}}text {}left [3 {left (frac {x} {L} {L}right)}}right)} ^ {2}right)}right)left (frac {x} {L}right)}right)Right]
each corresponding respectively to the degrees of freedom \({w}_{1}\), \({\theta }_{{y}_{1}}\),,, \({w}_{2}\), and \({\theta }_{{y}_{2}}\) of the two nodes of the element,
It is easy to verify that \(({\xi }_{i}\text{,}{\xi }_{i+4})\) couples are verifying 4.3.1-2. In addition:
\[\begin{split}\ begin {array} {cc}
{({\ xi} _ {1}\ text {,} {,} {\ xi} _ {5})} _ {(0)} = (1\ text {,} 0) &\ text {} {} {({\ xi} _ {1} _ {1}\ text {,} _ {5})} _ {\ xi} _ {5})} _ {(L)} = (0\ text {,} 0) _ {,} 0)\\
{({\ xi} _ {2}\ text {,} {,} {\ xi} _ {6})} _ {(0)} = (0\ text {,} 1) &\ text {} {} {({\ xi} _ {2} _ {2}}\ 2}\\\\\
{({\ xi} _ {3}\ text {,} {,} {\ xi} _ {7})} _ {(0)} = (0\ text {,} 0) &\ text {} {} {({\ xi} _ {3} _ {3} _ {3})}\\
{({\ xi} _ {4}\ text {,} {,} {\ xi} _ {8})} _ {(0)} = (0\ text {,} 0) &\ text {} {} {({\ xi} _ {4} _ {4}}\ text {,} {4}}\ text {,} _ {4}}\ text {,} _ {8})} _ {(L)} = (0\ text {,} _ {,} 1)
\ end {array}\end{split}\]
The stiffness matrix is easily deduced from 4.3.1-3 (by ordering the columns following \((w(0)\text{,}{\theta }_{y}(0)\text{,}w(L)\text{,}{\theta }_{y}(L))\)).
\[\begin{split}K=\ frac {12 {\ mathit {EI}}} _ {y}} {{y}} {{L} ^ {3}\ left (1+ {\ varphi} _ {y}\ right)}\ left (
\ begin {array} {cccc}
1& -\ frac {L} {2} & -1& -\ frac {L} {2}\\
&\ frac {\ left (4+ {\ varphi} _ {y}\ right) {L} ^ {2}} {12} &\ frac {L} {2} &\ frac {\ left (2- {\ varphi} _ {\ varphi} _ {y}\ right) {L} ^ {2}} {2} {2}} {12}\\ left (2- {\ varphi} _ {y}\ right) {L} {2}} {2}} {12}\\ left (2- {\ varphi} _ {y}\ right) {L} {2}} {2} {2}} {2} {
&\ mathit {sym} & 1&\ frac {L} {2}\\
& & &\ frac {\ left (4+ {\ varphi} _ {y}\ right) {L} ^ {2}} {12}} {12}
\ end {array}
\ right)\end{split}\]
It is clear that the calculation of the efforts is carried out in the same way as in [§ 4.1.3].
4.3.2. In-plane bending (xOy)
The stiffness matrix for a flexural movement in plane \((\mathrm{xOy})\) is obtained in the same way as in the previous case. The test functions that lead to an exact expression of the nodal forces must this time verify (equation analogous to [eq]):
\[\]
: label: 4.3.2-1
- begin {cases}
Psi « -omega “=0\
{mathit {EI}} _ {z}omega « + {k} _ {y} SGleft (Psi “-omegaright) =0
end {cases}
The pairs of test functions chosen are:
\[\ left ({\ xi} _ {1}\ text {,}\ text {,}} - {\ xi}} _ {5}\ right)\ text {;}\ left (- {\ xi} _ {2}\ text {,} {\ xi}} _ {\ xi}} _ {6}\ right)\ text {;}\ left ({\ xi} _ {7}\ right)\ text {;}\ left (- {\ xi} _ {4}\ text {,} {\ xi} _ {8}\ right)\]
each corresponding respectively to the degrees of freedom \({v}_{1}\), \({\theta }_{{z}_{1}}\),,, \({v}_{2}\), and \({\theta }_{{z}_{2}}\) of the two nodes of the element,
The \({\xi }_{i}\) are given by [eq] by having replaced \({\phi }_{y}\) by \({\phi }_{z}=\frac{\text{12}{EI}_{z}}{{k}_{y}{\text{SGL}}^{2}}\). The stiffness matrix obtained is:
\[\begin{split}\ mathrm {K} =\ frac {12 {EI} _ {z} _ {z}} {{L} ^ {3}\ left (1+ {\ varphi} _ {z}\ right)}\ left (
\ begin {array} {cccc} 1&
\ frac {L} {2} & -1&\ frac {L} {2}\\
&\ frac {\ left (4+ {\ varphi} _ {z}\ right) {\ text {L}}} ^ {2}} {12} & -\ frac {L} {2} &\ frac {\ left (2- {\ varphi} (2- {\ varphi}} _ {z}\ right) {\ text {L}}} {2}} {2}} {12}\ left (2- {\ varphi} _ {z}\ right) {\ text {L}} {2}} {2} {2} {2} {2} {2} {2}} {12}\ left (2- {\ varphi}\ left (2- {\ varphi}} _ {z}\
&\ text {sym} & 1& -\ frac {L} {2}\\
& & &\ frac {\ left (4+ {\ varphi} _ {z}\ right) {\ text {L}} ^ {2}}} {12}} {12}
\ end {array}
\ right)\end{split}\]
4.3.3. Elbow modeling
The modeling of pipe elbows is modeled by right beam elements POU_D_T by discretizing the geometry of the elbow with 20 to 40 meshes for a 90° elbow.
In the case of modeling pipe bends (hollow circular section), it is possible to take into account the effect of ovalization on stiffness by weighting inertias \({I}_{y}\) and \({I}_{z}\) (see § 10) by flexibility coefficients given by the user. This coefficient is to be entered using the keyword POUTRE/SECTION =” COUDE “of the command AFFE_CARA_ELEM.
4.3.4. Determination of the mass matrix consistent with the stiffness matrix
CALC_MATR_ELEM operator “MASS_MECA” option.
4.3.4.1. In-plane bending (xOz)
Consider the bending motion in plane \((xOz)\), the work of inertial forces is written as:
\[W= {\ int} _ {o} ^ {L} (w {\ rho}} (w {\ rho}} _ {\ rho} _ {\ rho} _ {{I} _ {z}}}\ ddot {\ rho}}}\ ddot {\ rho}}}}\ ddot {\ rho}}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} _ {z}}}\ ddot {\ theta} int} _ {S}\ rho\ mathrm {dS}\ text {and} {\ rho} _ {{I} _ {z}} = {\ int} _ {S}\ rho {y}}\ rho {y}} ^ {2}\ mathrm {dS}\]
In the case of a homogeneous material, we have: \({\rho }_{m}=\rho S\text{}\) and \({\rho }_{{I}_{z}}=\rho {I}_{z}\)
\(w(x,t)\) and \({\theta }_{y}(x,t)\) are discretized on the basis of the test functions introduced for the calculation of the stiffness matrix, i.e.:
\[\]
- begin {cases}
wleft (text {x,} tright) = {xi} _ {1}left (xright) {text {w}} _ {1}left (tright) + {xi} _ {2} _ {2}left (xright) {2}}left (xright) _ {right)text {+} {xi} _ {3}left (xright) {text {w}}} _ {2}left (tright) + {xi} _ {4}left (xright) {theta} {theta} _ {{y}} _ {y}} _ {2}}left (tright)\
{theta} _ {y}left (text {x,} tright) = {xi} _ {5}left (xright) {text {w}} _ {1}left (tright) + {xi} + {xi} _ {right) + {xi} _ {right) + {xi} _ {7}left (xright) {text {w}}} _ {2}left (tright) + {xi} _ {8}left (xright) {theta} {theta}} _ {{y}} _ {2}}left (tright)
end {cases}
in other words: \(w={w}^{t}{\xi}_{w}\) with \(\underset{\text{~}}{\mathrm{w}}=\left(\begin{array}{c}{w}_{1}\\ {\theta }_{{y}_{1}}\\ {w}_{2}\\ {\theta }_{{y}_{2}}\end{array}\right)\) and \({\xi }_{\mathrm{w}}=\left(\begin{array}{c}{\xi }_{1}\\ {\xi }_{2}\\ {\xi }_{3}\\ {\xi }_{4}\end{array}\right)\), \({\theta }_{y}={\underset{\text{~}}{w}}^{t}{\xi }_{{\theta }_{y}}\) with \({\xi }_{{\theta }_{y}}=\left(\begin{array}{c}{\xi }_{5}\\ {\xi }_{6}\\ {\xi }_{7}\\ {\xi }_{8}\end{array}\right)\)
By integrating these notations into the expression of the work of inertial forces, we have:
\[W= {\ int} _ {0} ^ {L} {\ rho} _ {\ rho} _ {m} {\ underset {\ text {~}}} {\ mathrm {w}}} {\ mathrm {t}}} {\ xi} _ {\ mathrm {w}}}} ^ {\ mathrm {t}}}} {\ mathrm {t}}} {\ mathrm {t}}} {\ mathrm {t}}} {\ mathrm {t}}} {\ mathrm {t}}} {\ mathrm {t}}} {\ xi} _ {\ mathrm {w}}}}\ underset {\ text {~}} {\ ddot {\ mathrm {w}}}}} + {\ rho} _ {{I} _ {z}} {\ underset {\ text {~}} {\ mathrm {w}}}} {\ mathrm {w}}}}} {\ mathrm {w}}}}} {\ xi} _ {\ theta} _ {\ mathrm {y}}} {\ xi} _ {\ theta} _ {\ mathrm {y}}} {\ xi}} {\ xi} _ {\ theta}} {\ theta}} {\ mathrm {y}}} {\ xi}} _ {{\ theta} _ {\ mathrm {y}}} ^ {\ mathrm {t}}}}\ ddot {\ underset {\ text {~}}} {\ mathrm {w}}}} {\ mathrm {w}}}}\ mathit {xx}\]
From this we deduce the expression for the mass matrix:
\[ \begin{align}\begin{aligned}\begin{split}M & =\ left ({m} _ {\ text {ij}}\ right)\\\end{split}\\{\ text {m}} _ {\ text {ij}}} & = {\ int}}} & = {\ int} _ {o} ^ {L}\ rho\ text {S}\ left (x\ right) {\ xi}} & = {\ text {ij}} {j}\ left (x\ right) +\ rho {\ text {I}}} _ {z}\ left (x\ right) _ {z}\ left (x\ right) _ {z}\ left (x\ right)) {\ xi} _ {\ text {i+} 4}\ left (x\ right) {\ xi} _ {\ text {j+} 4}\ left (x\ right)\ text {xx}\end{aligned}\end{align} \]
for \(i\) from 1 to 4 and \(j\) from 1 to 4. Either:
\(\begin{array}{cc}\mathrm{M}& =\frac{\mathrm{\rho }S}{{\left(1+{\mathrm{\varphi }}_{y}\right)}^{2}}\left(\begin{array}{cccc}\frac{13L}{35}+\frac{7L{\mathrm{\varphi }}_{y}}{10}+\frac{L{\mathrm{\varphi }}_{y}^{2}}{3}& \frac{-11{L}^{2}}{210}-\frac{11{L}^{2}{\mathrm{\varphi }}_{y}}{120}-\frac{{L}^{2}{\mathrm{\varphi }}_{y}^{2}}{24}& \frac{9L}{70}+\frac{3L{\mathrm{\varphi }}_{y}}{10}+\frac{L{\mathrm{\varphi }}_{y}^{2}}{6}& \frac{13{L}^{2}}{420}+\frac{3{L}^{2}{\mathrm{\varphi }}_{y}}{40}+\frac{{L}^{2}{\mathrm{\varphi }}_{y}^{2}}{24}\\ \phantom{\rule{2em}{0ex}}& \frac{{L}^{3}}{105}+\frac{{L}^{3}{\mathrm{\varphi }}_{y}}{60}+\frac{{L}^{3}{\mathrm{\varphi }}_{y}^{2}}{120}& \frac{-13{L}^{2}}{420}-\frac{3{L}^{2}{\mathrm{\varphi }}_{y}}{40}-\frac{{L}^{2}{\mathrm{\varphi }}_{y}^{2}}{24}& \frac{-{L}^{3}}{140}-\frac{{L}^{3}{\mathrm{\varphi }}_{y}}{60}-\frac{{L}^{3}{\mathrm{\varphi }}_{y}^{2}}{120}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \frac{13L}{35}+\frac{7L{\mathrm{\varphi }}_{y}}{10}+\frac{L{\mathrm{\varphi }}_{y}^{2}}{3}& \frac{11{L}^{2}}{210}+\frac{11{L}^{2}{\mathrm{\varphi }}_{y}}{120}+\frac{{L}^{2}{\mathrm{\varphi }}_{y}^{2}}{24}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \frac{{L}^{3}}{105}+\frac{{L}^{3}{\mathrm{\varphi }}_{y}}{60}+\frac{{L}^{3}{\mathrm{\varphi }}_{y}^{2}}{120}\end{array}\right)\\ \phantom{\rule{2em}{0ex}}& +\frac{\mathrm{\rho }{I}_{z}}{{\left(1+{\mathrm{\varphi }}_{y}\right)}^{2}}\left(\begin{array}{cccc}\frac{6}{5L}& \frac{-1}{10}+\frac{{\mathrm{\varphi }}_{y}}{2}& \frac{-6}{5L}& \frac{-1}{10}+\frac{{\mathrm{\varphi }}_{y}}{2}\\ \phantom{\rule{2em}{0ex}}& \frac{2L}{15}+\frac{L{\mathrm{\varphi }}_{y}}{6}+\frac{L{\mathrm{\varphi }}_{y}^{2}}{3}& \frac{1}{10}-\frac{{\mathrm{\varphi }}_{y}}{2}& \frac{-L}{30}-\frac{L{\mathrm{\varphi }}_{y}}{6}+\frac{L{\mathrm{\varphi }}_{y}^{2}}{6}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \frac{6}{5L}& \frac{1}{10}-\frac{{\mathrm{\varphi }}_{y}}{2}\\ \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \phantom{\rule{2em}{0ex}}& \frac{2L}{15}+\frac{L{\mathrm{\varphi }}_{y}}{6}+\frac{L{\mathrm{\varphi }}_{y}^{2}}{3}\end{array}\right)\end{array}\)
It should be noted, as in [§ 4.1.4], that in the dynamic case, we are not guaranteed to have an exact solution at the nodes, as is the case in static.
4.3.4.2. Flexion movement around the axis (Oz)
Likewise, for the flexure movement around the \((Oz)\) axis, in the \((xOy)\) plane, the work of the inertial forces is written:
\[{\ int} _ {O} ^ {L} (v {\ rho}} _ {m}\ ddot {v} + {\ theta} _ {\ rho} _ {{I} _ {y}}} {\ ddot {\ theta}}} {\ ddot {\ theta}} _ {z}})\ mathrm {x}\ text {with} {\ rho} _ {y}} _ {y}} {\ y}} {\ ddot {\ theta}}} _ {y}} {\ ddot {\ theta}}} _ {z})\ mathrm {xx}\ text {with} {\ rho} _ {y}} _ {y}}} {\ ddot {\ theta}}}}} = {\ int} _ {S}\ rho {z} ^ {2}\ mathrm {dS} =\ rho {I} _ {y}\]
This time \(v(x,t)\) and \({\theta }_{z}(x,t)\) are discretized in accordance with [§ 4.3.2] by:
\[\begin{split}\ begin {cases}
v (\ text {x,} t) & = {\ xi} _ {1} _ {1} (x) {\ text {v}} _ {1} (t) - {\ xi} _ {2} (x) {\ theta} _ {\ theta} _ {1}} _ {1}} (t) {{z} _ {1}} (t) {\ text {v}}} _ {2} (x) {\ text {v}}} _ {2} (t) - {\ xi} _ {4} (x) {\ theta} _ {{z} _ {2}} (t)\\
{\ theta} _ {z} (\ text {x,} t) & =- {\ x} _ {5} (x) {\ text {v}} _ {1} (t) + {\ xi} _ {6} _ {6} (x) (x) {\ theta} (x) {\ theta} _ {1}} (t) - {\ xi} _ {7} (x) {6} (x) (x) {\ text {v}} _ {2} (t) + {\ xi} _ {8} (x) {\ theta} _ {{z} _ {2}}} (t)
\ end {cases}\end{split}\]
We then get the following mass matrix:
\(\begin{array}{cc}\mathrm{M}& =\frac{\rho S}{{\left(1+{\varphi }_{z}\right)}^{2}}\left(\begin{array}{cccc}\frac{13L}{35}+\frac{7L{\varphi }_{z}}{10}+\frac{L{\varphi }_{z}^{2}}{3}& \frac{11{L}^{2}}{210}+\frac{11{L}^{2}{\varphi }_{z}}{120}+\frac{{L}^{2}{\varphi }_{z}^{2}}{24}& \frac{9L}{70}+\frac{3L{\varphi }_{z}}{10}+\frac{L{\varphi }_{z}^{2}}{6}& \frac{-13{L}^{2}}{420}-\frac{3{L}^{2}{\varphi }_{z}}{40}-\frac{{L}^{2}{\varphi }_{z}^{2}}{24}\\ & \frac{{L}^{3}}{105}+\frac{{L}^{3}{\varphi }_{z}}{60}+\frac{{L}^{3}{\varphi }_{z}^{2}}{120}& \frac{13{L}^{2}}{420}+\frac{3{L}^{2}{\varphi }_{z}}{40}+\frac{{L}^{2}{\varphi }_{z}^{2}}{24}& \frac{-{L}^{3}}{140}-\frac{{L}^{3}{\varphi }_{z}}{60}-\frac{{L}^{3}{\varphi }_{z}^{2}}{120}\\ & & \frac{13L}{35}+\frac{7L{\varphi }_{z}}{10}+\frac{L{\varphi }_{z}^{2}}{3}& \frac{-11{L}^{2}}{210}-\frac{11{L}^{2}{\varphi }_{z}}{120}-\frac{{L}^{2}{\varphi }_{z}^{2}}{24}\\ & & & \frac{{L}^{3}}{105}+\frac{{L}^{3}{\varphi }_{z}}{60}+\frac{{L}^{3}{\varphi }_{z}^{2}}{120}\end{array}\right)\\ & +\frac{\rho {I}_{y}}{{\left(1+{\varphi }_{z}\right)}^{2}}\left(\begin{array}{cccc}\frac{6}{5L}& \frac{1}{10}-\frac{{\varphi }_{z}}{2}& \frac{-6}{5L}& \frac{1}{10}-\frac{{\varphi }_{z}}{2}\\ & \frac{2L}{15}+\frac{L{\varphi }_{z}}{6}+\frac{L{\varphi }_{z}^{2}}{3}& \frac{-1}{10}+\frac{{\varphi }_{z}}{2}& \frac{-L}{30}-\frac{L{\varphi }_{z}}{6}+\frac{L{\varphi }_{z}^{2}}{6}\\ & & \frac{6}{5L}& \frac{-1}{10}+\frac{{\varphi }_{z}}{2}\\ & & & \frac{2L}{15}+\frac{L{\varphi }_{z}}{6}+\frac{L{\varphi }_{z}^{2}}{3}\end{array}\right)\end{array}\)
In the Euler-Bernoulli beam model, the effects of transverse shear are neglected. To obtain the mass and stiffness matrices associated with this model, it is therefore sufficient to cancel the variables \({\phi }_{y}\) and \({\phi }_{z}\) contained in the mass and stiffness matrices of the Timoshenko model. (\({\phi }_{y}\) and \({\phi }_{z}\) involve the shape coefficients \({k}_{y}\) and \({k}_{z}\), which are the opposite of the shear coefficients \({A}_{Y}\) and \({A}_{Z}\)).
It should be noted that in the Euler-Bernoulli model programmed in Aster, rotational inertia is also neglected. For this model, it is therefore necessary to cancel the terms in \(\rho {I}_{z}\) and \(\rho {I}_{y}\) in the mass matrix of the Timoshenko model.
4.4. Reduced mass matrix using the concentrated mass technique
The mass matrix is thus reduced to a diagonal matrix and is obtained by the option “MASS_MEGA_DIAG” of the CALC_MATR_ELEM operator.
The beam element is considered to have a constant cross section \(S\) and a constant density \(\rho\).
The so-called « Lumping » technique consists in adding up on the diagonal all the terms of the row of the coherent matrix and cancelling all the extra-diagonal terms.
For the diagonal component related to the pull-compression movement \(({M}_{11})\) and that related to the torsional movement \(({M}_{44})\), we have:
\[\]
- begin {array} {}
{M} _ {text {11}} &=rhotext {S}frac {L} {2}\
{M} _ {text {44}} &=rho ({I} _ {y} {text {+I}} _ {z})frac {L} {2})frac {L} {2}
end {array}
with \({I}_{y},{I}_{z}\): geometric moments
It can be considered that these components were obtained by dividing the beam element into two equal parts of length \(\frac{L}{2}\) and then combining the mass and inertia obtained at the node of the half-element. For, \({M}_{44}\) the previous expression corresponds to a choice: we could also have written: \({M}_{\text{44}}=\rho C\frac{L}{2}\).
Note
Comparison with numerical integration methods
It can be noted that if we perform an approximate integration in the following way:
\({\int }_{e}f=\sum _{\text{i=1,n}}\frac{\text{mes}(e)}{n}\text{f}({a}_{i}^{e})\)
(\({a}_{i}^{e}\): \(i\) nodes of the element \(e\), \(n\): number of nodes of the element)
we get an identical result (for a beam: * \(\mathrm{mes}(e)=L\) and \(n=2\)).
The diagonal components related to the flexure movements that are programmed are:
\[\begin{split}{M} _ {\ text {22}} &=\ mathrm {\ rho}\ text {S}\ frac {L} {2}\\
{M} _ {\ text {33}} &=\ mathrm {\ rho}\ text {S}\ frac {L} {2}\\
{M} _ {\ text {55}} &=\ mathrm {\ rho}\ text {S}\ frac {{L} ^ {3}} {\ text {105}}} +\ rho {\ text {I}}} +\ rho {\ text {I}}}\\
{M} _ {\ text {66}} &=\ mathrm {\ rho}\ text {\ rho}\ text {S}\ frac {{L} ^ {3}} {\ text {105}}} +\ rho {\ text {I}}} +\ rho {\ text {I}}} +\ rho {\ text {I}}}} +\ rho {\ text {I}}}} _ {z}\ frac {2L} {\ text {105}}} +\ rho {\ text {105}}} +\ rho {\ text {I}}}\end{split}\]
We find the components \({M}_{22}\) and \({M}_{33}\) linked to the translations of flexural movements by the technique of masses concentrated at the nodes.
4.5. Centrifugal stiffness matrix
The centrifugal stiffness matrix results from a rotational load characterized by the speed vector \(\omega ={}^{t}(\begin{array}{ccc}{\omega }_{x}& {\omega }_{y}& {\omega }_{z}\end{array})\). Let’s say \(u={}^{t}(\begin{array}{ccc}{u}_{x}& {u}_{y}& {u}_{z}\end{array})\) the displacement vector.
Her expression is \({K}_{\omega }(u)=-(\frac{1}{2}){\int }_{V}\rho ((\omega \wedge u)\cdot (\omega \wedge u))\mathrm{dV}\)
We can write: \({K}_{\omega }(u)=-(\frac{1}{2}){\int }_{V}\rho (({u}^{t}{\Omega }^{t})\cdot (\Omega u))\mathrm{dV}\) with \(\Omega =\left(\begin{array}{ccc}0& -{\omega }_{z}& {\omega }_{y}\\ {\omega }_{z}& 0& -{\omega }_{x}\\ -{\omega }_{y}& {\omega }_{x}& 0\end{array}\right)\)
By decomposing the displacements according to the shape functions \(({N}_{i},i=\mathrm{1,2}),({\xi }_{i}^{y},{\xi }_{i}^{z},i=\mathrm{1,4})\) defined above, it comes: \(u=\mathrm{Nq}\) where \(q\) designates all the degrees of freedom of the 2 nodes of the beam. On an element \({V}_{e}\) the elementary matrix \({K}_{e}\) is written:
\[{K} _ {e} =- (\ frac {1} {2}) {\ frac {1} {2}) {\ int} _ {e}}}\ rho (({N} ^ {t}} {\ omega} ^ {t})\ cdot (\ omega N)) {\ mathrm {dV}}} _ {e}\]
To have a centrifugal stiffness matrix consistent with the stiffness matrix and the coherent mass matrix, u is discretized on the basis of the test functions introduced for the calculation of the stiffness matrix, i.e.:
\[{u} _ {x} (\ text {x,} t) = {N} _ {1} (x) {\ text {u}} _ {1} (t)\ text {+} {N} {N} _ {2} _ {2} (x) {\ text {u}} _ {2} (t)\]
\[{u} _ {y} (\ text {x,} t) = {\ xi} _ {1} ^ {z} (x) {\ text {v}} _ {1} (t) - {\ xi} _ {2} _ {2} ^ {z} (t) {\ z} (x) {\ theta} _ {z} (x) _ {1}} (t)\ text {+} {\ xi} _ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ (x) {\ z} (x) {\ theta} _ {1}} (t)\ text {+} {\ xi} _ {3} ^ {3} ^ {z} (x) {\ text {v}} _ {2} (t) - {\ xi} _ {4} ^ {z} (x) {\ theta} _ {{z}} _ {2}} (t)\]
\[{u} _ {z} (\ text {x,} t) = {\ xi} _ {1} ^ {y} (x) {\ text {w}} _ {1} (t) + {\ xi} _ {2} _ {2} ^ {y} (x) {\ y} (x) {\ theta} _ {\ theta} _ {1}}} (t)\ text {+} {\ xi} _ {2} ^ {2} ^ {2} ^ {2} ^ {2} ^ (x) {\ theta} (x) {\ theta} _ {1}}} (t)\ text {+} {\ xi} _ {3} ^ {3} (x) {\ theta} _ {3} (x) {\ theta} _ {1}}} y} (x) {\ text {w}} _ {2} (t) + {\ xi} _ {4} ^ {y} (x) {\ theta} _ {{y} _ {2}}} (t)\]
The matrix \(N\) is easily deduced from this:
\[\begin{split}N=\ left (\ begin {array} {cccccccccccc}
{N} _ {1} & 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0\\
0& {\ xi} _ {1} ^ {z} & 0& 0& 0& 0& 0& - {\ xi} _ {2} ^ {z} & 0& {\ xi} _ {3} ^ {z} & 0& 0& 0& 0& 0& 0& 0& - {\ xi} _ {4} ^ {z}\\
0& 0& {\ xi} _ {1} ^ {y} & 0& {\ xi} & 0& {\ xi} _ {2} ^ {y} & 0& 0& {\ xi} _ {3} ^ {y} _ {3} ^ {y} _ {3} ^ {y} _ {y} & 0
\ end {array}\ right)\end{split}\]
Since the matrices are sparse, we can easily explain the product \(({N}^{t}{\Omega }^{t})\cdot (\Omega N)\)
It is a symmetric \((12,12)\) matrix made up of the following blocks:
\[\begin{split}{k} _ {e} =\ left (\ begin {array} {cc} {cc} {k} _ {11} &\ mathit {sym}\\ {k} _ {12} _ {12} & {k} _ {22}\ end {array}\ right)\end{split}\]
\[\begin{split}{k} _ {11} =\ left (((\ begin {array} {cccccc}} ({\ omega} _ {y} ^ {2} + {\ omega} _ {z} ^ {2}) {N} _ {1} {1}} {1}} ^ {2} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} & thrm {sym}\\ - {\ omega} _ {\ omega} _ {x} {\ omega} _ {y} {N} _ {1} ^ {z} & ({\ omega} _ {x} ^ {2} ^ {2} + {\ omega} ^ {2}} + {\ omega} ^ {2}) {\ omega} _ {z} _ {1} ^ {z}) &\ mathrm m {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym}\\ - {\ omega} _ {z} {N} _ {1} {\ xi} _ {1} {xi} _ {\ xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} ^ {y} _ {1} ^ {y} & - {\ omega} _ {z} {\ xi} _ {1} {xi} _ {1} {xi} _ {1} {xi} _ {1} {xi}} {\ xi} _ {1} ^ {z} & ({\ omega} _ {x} _ {x} ^ {2} + {\ omega} _ {y} ^ {2}) {({\ xi} _ {1} ^ {y})} {1} ^ {y})}} ^ {y})}} ^ {y})}} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y} omega} _ {x} ^ {2} {2} {N} _ {1} ^ {2} &\ mathrm {sym} &\ mathrm {sym}\\ - {\ omega} _ {x} {\ omega} _ {z} {N} _ {1} {\ xi} _ {2} _ {2} ^ {y}} & - {\ y}} & - {\ omega} _ {y} {\ omega} _ {z} {\ xi} _ {2} _ {1} {z} {z} _ {z} {z} {2} _ {2} {\ xi} _ {2}} ^ {y} & ({\ omega} _ {x} _ {x} ^ {2} + {\ omega} _ {y} ^ {2}) {\ xi} _ {1} ^ {xi} _ {2} ^ {y} _ {2} _ {2} _ {2} & 0& ({\ omega} _ {x} ^ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {\ xi} _ {2} _ 2} ^ {y})} ^ {2} &\ mathrm {2} &\ mathrm {sym}\\ sym}\\ {\ omega} _ {1} {\ xi} _ {2} ^ {z} & - ({\ omega} ^ {z} & - ({\ omega} _ {x} ^ {2}) {\ xi} ^ {2}) {\ xi} _ {z} _ {2} _ {2}) {2} _ {2} _ {2}) {\ xi} ^ {z} _ {z} & - ({\ omega} _ {x} ^ {2}) {\ xi} ^ {2} _ {2}) {2} _ {2} _ {2}) {\ xi} ^ {z} _ {z} _ {z}} {z} {\ xi} _ {2} ^ {z} & {\ omega} & {\ omega} _ {y} {\ omega} _ {z} {\ xi} _ {2} {\ xi} _ {2} ^ {z} _ {2} ^ {z}} {z} {z} {z} {\ xi} _ {2} ^ {y} {\ xi} _ {2} {y} {\ xi} _ {2} {y} {\ xi} _ {2} ^ {z} & ({\ omega} _ {x}} ^ {2} + {\ omega} _ {z} ^ {2}) {({\ xi} _ {2} ^ {z})} ^ {2}\ end {array}\ right)\end{split}\]
\[\begin{split}{k} _ {22} =\ left (((\ begin {array} {cccccc}} ({\ omega} _ {y} ^ {2} + {\ omega} _ {z} ^ {2}) {N} _ {2}) {N} _ {2}) {2}} {2} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm thrm {sym}\\ - {\ omega} _ {\ omega} _ {x} {\ omega} _ {y} {N} _ {2} {\ xi} _ {3} ^ {z} & ({\ omega} _ {x}} ^ {2} ^ {2}) {\ omega} _ {z} _ {3} ^ {z}) &\ mathrm m {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym}\\ - {\ omega} _ {z} {N} _ {2} {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {\ xi} _ {2} {\ xi} _ {} {\ xi} _ {3} ^ {z} & ({\ omega} _ {x} _ {x} ^ {2} + {\ omega} _ {y} ^ {2}) {({\ xi} _ {3} ^ {y})} {3} ^ {y})}} ^ {y})}} ^ {y})}} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y})} ^ {y} omega} _ {x} ^ {2} {2} {N} _ {2} ^ {2} &\ mathrm {sym} &\ mathrm {sym} &\ mathrm {sym}\\\ - {\ omega} _ {z} {N} _ {2} {\ xi} _ {4} ^ {y} _ {4} ^ {y}} & - {\ y}} & - {\ omega} _ {y} {\ omega} _ {\ xi} _ {4} {\ xi} _ {4} {\ xi} _ {4} ^ {x} _ {4} ^ {y}} & - {\ omega} _ {\ omega} _ {3} {\ xi} _ {4} {z} {\ xi} _ {4} {\ xi} _ {4} ^ {x}} _ {4} ^ {y}} {y}} _ {4}} ^ {y} & ({\ omega} _ {x} _ {x} ^ {2} + {\ omega} _ {y} ^ {2}) {\ xi} _ {3} ^ {xi} _ {4} ^ {y} _ {4} ^ {y} & 0& ({\ omega} _ {x} ^ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {({\ xi}} _ {2}) {4} ^ {y})} ^ {2} &\ mathrm {2} &\ mathrm {sym}\\ sym}\\ {\ omega} _ {2} {\ xi} _ {4} ^ {z} & - ({\ omega} ^ {z} & - ({\ omega} _ {x} ^ {2}) {\ xi} ^ {2} _ {4} ^ {z} _ {z} _ {z} _ {z} & - ({\ omega} _ {x} ^ {2}) {\ xi} _ {z} _ {z} _ {z}} _ {z} _ {3}} & - ({\ omega} _ {x} ^ {2}) {\ xi} _ {z} _ {2}) {4} ^ {z} _ {z} {z} {\ xi} _ {4} ^ {z} & {\ omega} & {\ omega} _ {y} {\ omega} _ {z} {\ xi} _ {4} {\ xi} _ {4} ^ {z} _ {4} ^ {z}} {z} {z} {z} {\ xi} _ {4} ^ {y} {\ xi} _ {4} _ {4} {y} {\ xi} _ {4} ^ {z} & ({\ omega} _ {x}} ^ {2} + {\ omega} _ {z} ^ {2}) {({\ xi} _ {4} ^ {z})} ^ {2}\ end {array}\ right)\end{split}\]
\[\begin{split}{k} _ {12} =\ left (\ begin {array} {cccccc} ({\ omega} _ {y} ^ {2} + {\ omega} _ {z} ^ {2}) {N} _ {1} _ {1} {1} {N} {N} {1}) {1} {1} {1} {1} ^ {z} & - {\ omega} _ {x} {\ omega} _ {x} {\ omega} _ {z} {\ omega} _ {1} ^ {y} & 0& - {\ omega} _ {x} {\ omega} _ {\ omega} _ {\ omega} _ {z} {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {z} {N} _ {2} {\ xi} _ {2} ^ {z} _ {2} _ {\ omega} _ {y} {N} _ {1} {\ xi} _ {3} {\ xi} _ {3} ^ {z} _ {z} & ({\ omega} _ {x} ^ {2}) {\ xi} _ {2} _ {1}) {\ xi} _ {1} ^ {z} {\ xi} _ {3} ^ {z} & - {\ omega} & - {\ omega} _ {y} {\ omega} _ {1} ^ {y} {\ xi} _ {3} ^ {z} _ {3} ^ {z} & 0& - {\ z} & 0& - {\ omega} _ {\ omega} _ {z} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {2} {\ xi} _ {3} ^ {3} ^ {z}} _ {3} ^ {z}} {z} {x} _ {3} ^ {z}} _ {3} ^ {z}} {z} {x} _ {3} ^ {z}} _ {3} ^ {z}} {z}} {3} ^ {z} & - ({\ omega} _ {x} ^ {2} + {\ omega} _ {z} ^ {2}) {\ omega} ^ {2}) {\ xi} _ {2} ^ {z} {\ xi} _ {3} ^ {x} _ {\ xi} _ {\ xi} _ {\ omega} _ {z} {N} _ {1} {\ xi} _ {3} _ {3} ^ {y} _ {y}} & - {\ omega} _ {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y} {y}} ^ {z} & ({\ omega} _ {x} _ {x} ^ {2} + {\ omega} _ {y} ^ {2}) {\ xi} _ {\ xi} _ {3} ^ {y} _ {3} ^ {y} & 0& ({\ omega} ^ {2} & 0& ({\ omega} _ {x} ^ {2}) {\ xi} _ {2}) {\ xi} _ {2}) {\ xi} _ {2}) {\ xi} _ {2}) {\ xi} _ {2} ^ {y} {\ xi} _ {3} ^ {y} & {\ omega} & {\ omega} _ {y} {\ omega} _ {z} {\ xi} _ {\ xi} _ {2} _ {2} ^ {2} ^ {z} ^ {z} ^ {z}}\ {z}\ {z}\ {z}\ 0& 0& 0& 0 _ {1} {N} {N} _ {2} _ {2} ^ {2} & 0\\ - {\ omega} _ {x} {\ omega} _ {z} {\ omega} _ {z} {\ omega} _ {x} {\ omega} _ {y} {\ omega} _ {\ omega} _ {z} {\ omega} _ {z} {\ omega} _ {z} {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {x} {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {{\ omega} _ {y} ^ {2}) {\ xi}) {\ xi} _ {1} ^ {y} {\ xi} _ {4} ^ {y} & 0& ({\ omega} _ {x} ^ {2} + {\ omega} _ {y} ^ {2}) {\ xi}) {\ xi} _ {\ xi} _ {4} ^ {y} & {\ omega} _ {y} {\ omega} _ {\ omega} _ {z} {\ omega} _ {z} {\ xi} _ {2} ^ {z} {\ xi} _ {2} ^ {x} {\ omega} _ {x} {\ omega} _ {x} {\ omega} _ {x} {\ omega} _ {x} {\ omega} _ {y} {N} _ {1} {\ xi} _ {4} ^ {z} _ {4} ^ {z} & - ({\ omega} _ {z} ^ {2}) {\ xi} _ {\ xi} _ {1}} _ {1} ^ {z} _ {1} ^ {z} _ {4} ^ {z} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {z} _ {z} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {\ omega} _ {1} ^ {y} {\ xi} _ {4} ^ {z} _ {4} ^ {z} & 0& {\ omega} _ {z} {\ xi} _ {2} ^ {y} {\ xi} _ {\ xi} _ {4} {xi} _ {4} ^ {xi} _ {z}} {\ xi} _ {z} {\ xi} _ {z} {2}) {\ xi} _ {z} {2} _ {z} {\ xi} _ {z} {2}) {\ xi} _ {z} {2}) {\ xi} _ {z} {2}) {\ xi} _ {z} {2}) {\ xi} _ {z} {2}) {\ xi} _ {z} {2}) {\ xi} _ {2} ^ {z} {\ xi} _ {4} ^ {z}\ end {array}\ right)\end{split}\]
It remains to integrate this matrix on a beam element of length L.
Since the coefficients \({\phi }_{y}\) and \({\phi }_{z}\) are defined in 4.3.1 and 4.3.2, for \(k=y,z\), we have:
\[{\ int} _ {0} ^ {L} {N} _ {1} {\ varphi} _ {1} ^ {k}\ mathit {xx} =\ frac {L} {60}\ left (9+\ frac {{\ varphi}} _ {\ varphi}} _ {60}}\ left (9+\ frac {\ varphi} {60}}\ left (9+\ frac {\ varphi} {60}}\ left (9+\ frac {\ varphi} {60}}\ left (9+\ frac {\ varphi}} {60}}\ left (9+\ frac {\ varphi}} {60}\ left)\]
\[{\ int} _ {0} ^ {L} {N} _ {N} _ {2} {\ xi} _ {1} ^ {k}\ mathit {xx} =\ frac {L} {60}\ left (21-\ frac {{\ varphi}} _ {\ varphi}} _ {60}}\ left (21-\ frac {\ varphi} {60}}\ left (21-\ frac {\ varphi}} {60}\ left (21-\ frac {\ varphi}} {60}\ left (21-\ frac {\ varphi}}}\ right)\]
\[{\ int} _ {0} ^ {L} {N} {N} _ {1} {\ xi} _ {2} ^ {k}\ mathit {xx} =\ frac {- {L} ^ {2}} {2}} {120}} {120}} {120}}\ left (4+\ frac {\ varphi}} {120}} {120}}\ left (4+\ frac {\ varphi}} _ {k})} {120}}\ left (4+\ frac {\ varphi} _ {k})} {120}}\ left (4+\ frac {\ varphi}} {\]
\[{\ int} _ {0} ^ {L} {N} {N} _ {2} {\ xi} _ {2} ^ {k}\ mathit {xx} =\ frac {- {L} ^ {2}} {2}} {120}} {120}} {120}}\ left (6-\ frac {\ varphi}} {120}}\ left (6-\ frac {\ varphi}} {120}}\ left (6-\ frac {\ varphi}} _ {k})} {120}}\ left (6-\ frac {\ varphi}} {120}}\ left (6-\ frac {\ varphi}}} {120}}\ left (6-\ frac {\ varphi}}} {\]
\[{\ int} _ {0} ^ {L} {N} {N} _ {1} {\ xi} _ {3} ^ {k}\ mathit {xx} =\ frac {L} {60}\ left (20+\ frac {1}} _ {60}}\ left (20+\ frac {1}} {1}} {1}} {1} {1} {1} {(1+\ varphi} _ {k})}\ right)\]
\[{\ int} _ {0} ^ {L} {N} {N} _ {2} {\ xi} _ {3} ^ {k}\ mathit {xx} =\ frac {L} {60}\ left (10-\ frac {1}} _ {60}}\ left (10-\ frac {1}} {1}} {1}} {1} {1} {1} {(1+ {\ varphi} _ {k})}\ right)\]
\[{\ int} _ {0} ^ {L} {N} {N} _ {1} {\ xi} _ {\ mathrm {4çk}}}\ mathit {xx} =\ frac {{L} ^ {2}} {2}} {2}}} {120}} {120}\ left (5+\ frac {1}} {1} {(1+\ varphi} _ {k})}\ right)\]
\[{\ int} _ {0} ^ {L} {N} {N} _ {2} {\ xi} _ {4} ^ {k}\ mathit {xx} =\ frac {{L} ^ {2}} {2}} {120}} {120}\ left (5-\ text {120}} {120}}\ left (5-\ text {}}} {120}}\ left (5-\ text {}} {120}}\ left (5-\ text {}} {120}}\ left (5-\ text {}}} {120}}\ left (5-\ text {}} {120}}\ left)\]
Note that integrals \({\int }_{0}^{L}{({\xi }_{i}^{k})}^{2}\mathit{dx},k=y,z\) are derived from integrals \({\int }_{0}^{L}{\xi }_{i}^{y}{\xi }_{i}^{z}\mathit{dx}\) by making \(y=z\).
It therefore remains to calculate the 16 integrals \({\int }_{0}^{L}{\xi }_{i}^{y}{\xi }_{j}^{z}\mathrm{dx}\), which are not a priory symmetric in y and z (in fact, some are).
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}} {\ int}}} {\ xi} _ {1} ^ {z}\ mathit {xx} =\ frac {L} {L} {L} {{L} {{L} {{L} {{L}} {{L} {{L} {{L} {{L}} {(1)} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {+\ frac {7} {20} ({\ varphi} _ {y} _ {y} + {\ varphi} _ {z}) +\ frac {({\ varphi} _ {y} {\ varphi} {\ varphi} _ {y}} _ {y}} _ {z})} {3}\ right)\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}}} {\ int}}} {\ xi} _ {2} ^ {z}\ mathit {xx} =\ frac {{L} {L} {\ l} ^ {2}}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {{1} {105} +\ frac {1} {120} ({\ varphi} _ {y} + {\ varphi} _ {z}) +\ frac {({\ varphi} _ {\ varphi} _ {y} {y} {\ varphi} {\ varphi} _ {z})} {120}\ right)\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {3} ^ {x} _ {3} ^ {z}\ mathrm {xx} = {\ int} _ {0} ^ {0} ^ {0} ^ {0} ^ {0} ^ {} ^ {} ^ {x} ^ {x} = {0} ^ {x} ^ {x} = {\ int} _ {0} ^ {0} ^ {} ^ {0} ^ {X} ^ {X} ^ {X} ^ {X} ^ {X} ^ {X}\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {4} ^ {y} {\ xi} _ {4} ^ {z}\ mathit {xx} = {\ int} _ {0} ^ {0} ^ {L} ^ {L} ^ {x} ^ {L} {\ xi} _ {2} ^ {z} = {\ int} _ {0} ^ {x} ^ {x} = {0} ^ {x} ^ {L} ^ {L} ^ {L} {L} {\ xi} _ {2} ^ {z}\ mathit {xx}}\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}}} {\ xi} _ {2} ^ {z}\ mathit {xx} =\ frac {- {L} {- {L} {\ L}} {\ int}}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int ac {11} {210} +\ frac {(10 {\ varphi}} _ {y} +12 {\ varphi} _ {z})} {240} +\ frac {({\ varphi} _ {\ varphi} _ {y} {y} {\ varphi} _ {y}} {\ varphi} _ {z})} {24}\ right)\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}}} {\ xi} _ {1} ^ {z}\ mathit {xx} =\ frac {- {L} {- {L} {\ L}} {\ int}}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int ac {11} {210} +\ frac {(12 {\ varphi}} _ {y} +10 {\ varphi} _ {z})} {240} +\ frac {({\ varphi} _ {\ varphi} _ {y} {y} {\ varphi} _ {y}} {\ varphi} _ {z})} {24}\ right)\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}}} {\ xi} _ {3} ^ {z} =\ frac {L} {L} {(1+ {\ varphi}}} {\ int}}} {\ int}}} {\ int}} {\ int}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {+\ frac {3} {20} ({\ varphi} _ {y} _ {y} + {\ varphi} _ {z}) +\ frac {({\ varphi} _ {y} {\ varphi} {\ varphi} _ {y}} _ {y}} _ {z})} {6}\ right)\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {3} ^ {x} _ {1} ^ {z}\ mathit {xx} = {\ int} _ {0} ^ {L} ^ {L} ^ {L} {\ xi} _ {1} ^ {x} _ {3} ^ {z} = {\ int} _ {0} ^ {x} = {0} ^ {X} ^ {L} ^ {L} ^ {L} {L} {\ xi} _ {1} ^ {z}\ mathit {xx}\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {1} ^ {x} _ {4} ^ {z}\ mathit {xx} =- {\ int} _ {0} ^ {0} ^ {L} ^ {L} ^ {xi} _ {2} ^ {x} _ {3} ^ {z} =- {\ int} _ {0}} _ {0} ^ {x} ^ {z} _ {0} ^ {x} ^ {x} _ {0} ^ {x} ^ {x} ^ {x}\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {4} ^ {x} _ {1} ^ {z}\ mathit {xx} =- {\ int} _ {0} ^ {0} ^ {L} ^ {L} ^ {x} ^ {L} ^ {x} _ {3} ^ {x} _ {2} ^ {z} =- {\ int} _ {0} ^ {x} ^ {z} _ {0} ^ {x} ^ {x} _ {0} ^ {x} ^ {x}\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}}} {\ xi} _ {3} ^ {z}\ mathit {xx} =\ frac {- {L} {- {L} {\ L}} {\ int}}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int ac {13} {420} +\ frac {(5 {\ varphi} _ {y} +4 {\ varphi} _ {z})} {120} +\ frac {({\ varphi} _ {\ varphi} _ {y} {y} {y} {\ varphi} _ {y}} {\ varphi} _ {z})} {24}\ right)\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}}} {\ xi} _ {2} ^ {z}\ mathit {xx} =\ frac {- {L} {- {L} {\ L}} {\ int}}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int ac {13} {420} +\ frac {(4 {\ varphi} _ {y} +5 {\ varphi} _ {z})} {120} +\ frac {({\ varphi} _ {\ varphi} _ {y} {y} {y} {\ varphi} _ {z})} {24}\ right)\]
\[\ underset {0} {\ overset {L} {\ int}} {\ int}} {\ int}}} {\ xi} _ {4} ^ {z}\ mathit {xx} =\ frac {- {L} {- {L} {\ L}} {\ int}}} {\ int}}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int}} {\ int ac {1} {140} +\ frac {({\ varphi} _ {y} _ {y} + {\ varphi} _ {z} + {\ varphi} _ {\ varphi} _ {varphi} _ {z})} {120}\ right)\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {4} ^ {x} _ {2} ^ {z}\ mathit {xx} = {\ int} _ {0} ^ {L} ^ {L} ^ {L} {\ xi} _ {2} ^ {x} _ {4} ^ {z} = {\ int} _ {0} ^ {x} = {0} ^ {x} ^ {L} ^ {L} ^ {L} {L} {\ xi} _ {2} ^ {z}\ mathit {xx}\]
\[{\ int} _ {0} ^ {L} {\ xi} _ {\ xi} _ {3} ^ {x} _ {4} ^ {z}\ mathit {xx} =- {\ int} _ {0} ^ {0} ^ {L} ^ {L} ^ {xi} _ {1} ^ {x} _ {2} ^ {z} =- {\ int} _ {0}} ^ {x} _ {0} ^ {x} ^ {z} _ {0} ^ {x} ^ {x} _ {0} ^ {x} ^ {x} _ {0} ^ {x} ^ {x}\]
\[{\ int} _ {0} ^ {L} {\ x} {\ xi} _ {4} ^ {x} _ {3} ^ {z}\ mathit {xx} =- {\ int} _ {0} ^ {0} ^ {L} ^ {L} ^ {x} ^ {x} _ {2} ^ {x} _ {1} ^ {z} =- {\ int} _ {\ int} _ {0} ^ {x} ^ {x} _ {0} ^ {x} ^ {x} ^ {x}\]