.What are all the steps to calculate Von mises stress theory .
Step 1: Calculate the three principal stresses (σ1,σ2, and σ3) using principal stress equations or Mohr’s circle
method.,Step 2: Find out the Von Mises Stress (σv) following the equations mentioned above.Step 3: Determine the
value of the allowable stress (σy/N) of the material. σy is the Yield Strength and N is the factor of safety.Step 4:
Compare the value calculated in step 2 with the allowable value found in step 3. If the Value at step 2 is less than the
allowable value calculated at step 3, then the design is safe as per the Von Mises stress theory.
A rectangular plate of the sides ‘a’ and ‘b’ and total thickness ‘h’ as
shown in Figure 1 is considered.The plate consists of N number of
homogenous layers. All the layers are perfectly bounded together
and made up of linearly elastic and orthotropic material. The plate
occupies the region 0 ≤ x ≤ a, 0 ≤
≤ b h/2 ≤ ≤ h/2 in Cartesian coordinate s stem
Buckling terminology The topic of buckling is still unclear because the keywords of “stiffness”, “long” and
“slender” have not been quantified. Most of those concepts were developed historically from 1D studies. You need
to understand those terms even though finite element analysis lets you conduct buckling studies in 1D, 2D, and
3D. For a material, stiffness refers to either its elastic modulus, E, or to its shear modulus, G = E / (2 + 2 v) where v
is Poisson’s ratio.
In the past, numerous works have been carried out on the static properties of fiber reinforced polymer composites
(FRPCs). However, the dynamic properties of polymer matrix composites (PMCs) have not been fully investigated
for structural engineering applications. Determination of natural frequencies and mode shapes of p olymer matrix
composites has gained a lot of importance recently as it can help predict the possibility of resonance so that
measures can be taken to avoid it or reduce the high amplitudes associated with it. The high resonant amplitudes are
a major cause for a number of failures occurring in the aerospace industry.
What do mean by Inter-laminar Stresses in composite material.A state of stress at a point is normally expressed
by the six stress components; namely, σx,σy, σz, τxy, τxz, and τyz. As far as a laminate is concerned, σx, σy, and
τxy are called as in-plane(normal and shear) stresses and σz, τxz, and τyz are out-of-plane stresses or inter-laminar
(normaland shear) stresses. In a fiber reinforced laminate, the load is transferred to adjacent layers bymeans of these
inter-laminar stresses.
Listthe various types of failure theories
1. Maximum-Stress Failure Theory:/3.Maximum-Strain Failure Theory:
2. Tsai-Hill Failure Theory:/4.Tsai-Wu Failure Theory:
Tsai-Hill Failure Theory:
This theory provides a single criterion to predict the failure of a lamina. It states that under
plane stress condition the failure will occur when the following inequality is satisfied.
How do you form mathematical formulation of Laminated Plate under Consideration?
Write down the Equations Of Motion for the principle of virtual work.
Which affect the laminate CTE:
Fiber and void volumes, layup angle, fabric skewness, stacking sequence, thermal cycling, temperature dependence,
moisture effects, and material viscoelasticity.
Thermal expansion data for several composite mat -,
Including generic epoxy resins, various graphite, boron, and glass fibers, and unidirectional and woven fabric
composites in an epoxy matrix, have been compiled into one comprehensive report.
Brief about the temperature dependent
This practice may precipitate significant errors in composite design because of the temperature dependence of the
thermal expansion behavior of composite materials. This temperature dependence is mainly caused by the
mechanical and physical changes in the resin system. For this reason, the CTE values should be obtained from
thermal expansion test data for the specific design temperature range.
Brief aboutt he effect of moisture on composite materials
Moisture causes swelling and plasticization of the resin system. The swelling phenomenon alters internal stresses,
thus causing a dimensional change in the laminate.
In the distortion energy theory, yielding occurs when the von Mises stress σ ′ is reached, or exceeded, by a state
of stress in the machine component .. A distortion-energy theory was prompted from the observation that ductile
materials stressed hydrostatically exhibited yield strengths greatly in excess of the values given by simple tension
tests. It was then postulated that yielding was related somehow to the angular distortion of the stressed element,
rather than that yielding was a simple tensile or compressive phenomenon. Nowadays, the distortion-energy theory
for ductile materials states that yielding occurs when the distortion strain energy per unit volume reaches or exceeds
the distortion strain energy per unit volume for yield in simple tension or compression of the same mate
b Buckling Analysis of Laminated Composite Plates
In this section, an analytical solution for the buckling analysis of plate is developed using Navier solution
technique. The governing equations of the plate in case of static buckling are obtained by discarding transverse
load (q) and inertia terms (I0, I1, I2, I3) from Eqs. (13) – (16). The in-plane compressive ( 00 0 and N ,N N xx yy
xy ) forces now represents loads instead of reaction forces, as there is no transverse load. The values of in-plane
compressive forces are taken as 0 N kN xx 1 0 , 0 N kN yy 2 0 and
0 0 Nxy . The governing equations for static buckling are as follows:
Bending Analysis of Laminated Composite Plates Following the Navier solution technique, the governing equations
of the simply supported laminated composite plates in case of bending analysis are obtained by discarding in-plane
compressive loads ( 000 N ,N ,N xx yy xy ) and
In this work, the natural frequencies and mode shapes of a number of cantilever glass fiber reinforced polymer
composites (GFRPCs) and carbon fiber reinforced polymer composites (CFRPCs) are numerically obtained using
the commercial finite element analysis software (ANSYS). The laminates under study include 8 ply cantilevered
plates having a plate aspect ratio of 2 and fiber volume fractions of 0.3, 0.4, 0.5, and 0.6. The finite element analysis
procedure is described. The natural frequencies and mode shapes calculated using ANSYS are first validated with
the results obtained from previous literature. The agreement between the two results is found to be excellent. The
effect of change in the matrix material, hybridization, and laminate stacking sequence on the natural frequencies and
mode shapes are also investigated. It is found that hybridization and orientation of the outermost layer has more
significant influence on the natural frequencies of the laminated composite plates compared to fiber volume fraction
and change in the matrix material.
The natural frequencies of CFRP composite plates are found to be higher than that of GFRP composites. S2
fiberglass/epoxy plates have higher natural frequencies in comparison to E-glass/epoxy plates. However, T300
carbon/epoxy plates and T800 carbon/epoxy plates have almost similar natural frequencies. • As the fiber volume
fraction is increased, the natural frequencies corresponding to a given mode also increases. This increase is marginal
for the first three modes but comparatively higher for higher modes. • The natural frequencies of laminates with
epoxy as the matrix material and vinyl ester as the matrix material are almost the same in case of GFRP composite
plates. However, CFRP composite plates with vinyl ester as matrix material show higher natural frequencies in
comparison to CFRP composite plates with Epoxy as the matrix material. • Hybrid E-glass T300 carbon/Epoxy
plates have natural frequencies higher than that of E-glass/epoxy plates, but lower than that of T300 carbon/epoxy
plates. Moreover, [0°c /45°g /-45°g /90°c ]s hybrid plates have higher natural frequencies compared to [0°g /45°c /-
45°c /90°g ]S hybrid plates.
What do mean by Inter-laminar Stresses in composite material.A state of stress at a point is normally expressed
by the six stress components; namely, σx,σy, σz, τxy, τxz, and τyz. As far as a laminate is concerned, σx, σy, and
τxy are called as in-plane(normal and shear) stresses and σz, τxz, and τyz are out-of-plane stresses or inter-laminar
(normaland shear) stresses. In a fiber reinforced laminate, the load is transferred to adjacent layers bymeans of these
inter-laminar stresses.
Listthe various types of failure theories
3. Maximum-Stress Failure Theory:/3.Maximum-Strain Failure Theory:
4. Tsai-Hill Failure Theory:/4.Tsai-Wu Failure Theory:
Tsai-Hill Failure Theory:
This theory provides a single criterion to predict the failure of a lamina. It states that under
plane stress condition the failure will occur when the following inequality is satisfied.
How do you form mathematical formulation of Laminated Plate under Consideration?
Write down the Equations Of Motion for the principle of virtual work.
Case 1:Given data:-The applied stresses,
σx = 50 MPa, (Tension)/σy = -25 MPa (Compression)/τxy = 50 MPa (Positive shear)//The allowable stresses,/σLU
= 750 MPa, (Tension)/σ’LU = 400 MPa (Compression)//σTU = 50 MPa, (Tension)/σ’TU = 100
4.1.What do mean by Inter-laminar Stresses in composite material.A state of stress at a point is normally
expressed by the six stress components; namely, σx,σy, σz, τxy, τxz, and τyz. As far as a laminate is concerned, σx,
σy, and τxy are called as in-plane(normal and shear) stresses and σz, τxz, and τyz are out-of-plane stresses or inter-
laminar (normaland shear) stresses. In a fiber reinforced laminate, the load is transferred to adjacent layers bymeans
of these inter-laminar stresses.
4.2. Listthe various types of failure theories
5. Maximum-Stress Failure Theory:/3.Maximum-Strain Failure Theory:
6. Tsai-Hill Failure Theory:/4.Tsai-Wu Failure Theory:
4.3 Tsai-Hill Failure Theory:
This theory provides a single criterion to predict the failure of a lamina. It states that under
plane stress condition the failure will occur when the following inequality is satisfied.
How do you form mathematical formulation of Laminated Plate under Consideration?
Write down the Equations Of Motion for the principle of virtual work.
Case 1:Given data:-The applied stresses,
σx = 50 MPa, (Tension)/σy = -25 MPa (Compression)/τxy = 50 MPa (Positive shear)//The allowable stresses,/σLU
= 750 MPa, (Tension)/σ’LU = 400 MPa (Compression)//σTU = 50 MPa, (Tension)/σ’TU = 100
Residual stresses can add up to the applied load and may critically degrade the strength and integrity of the
composite structure. These stresses are particularly tricky as they satisfy equilibrium and offer no external indication
of their presence.
which affect the laminate CTE : fiber and void volumes, layup angle, fabric skewness, stacking sequence, thermal
cycling, temperature dependence, moisture effects, and material viscoelasticity
Moisture Effects ;The dimensional stability nf composites is highly affected by exposure to com- plex hygrothermal
histories. Moisture causes swelling and plasticization of the
resin system. The swelling phenomenon alters internal stresses, thus causing a dimensional change in the laminate.
When structures, machines, and engineering details are in service, they are usually subjected to surface forces or
body forces (inertial, gravitational, or electromagnetic) that cause combined stresses in their elements. To
completely define an element, it is necessary to specify the components of the stress tensor represented by the array:
The thermo-mechanical coupling for laminated composite and sandwich beams. In this way, the temperature and the
displacements are written under the form of a sum of products of unidimensional polynomials of x and z. For the
mechanical unknowns, a piecewise fourth-order Lagrange polynomial of z is chosen as it is particularly suitable for
such laminated structures (see [19]). The temperature unknowns are also interpolated with the same order expansion.
As far as the variation with respect to the axial coordinate is concerned, a 1D three-node FE is employed. Using this
method, each unknown function of x is classically approximated using one degree of freedom (dof) per node of the
mesh and the LW unknown functions of z are global .
Residual stress measurements by applying HDM using strain gauges
Sicot et al. [18] investigated the effects of cooling conditions on the development of residual stresses in [08] and
[02/902]s Carbon/epoxy [CE] composite. The models proposed by So the and Lake [15] were used to determine the
residual stresses by employing the incremental hole drilling method (IHDM). Calibration coefficients required for
residual stress determination were obtained using finite element analysis (FEA). The stresses in [08] CE composite
laminas increased with the depth of increment and a stress increase across the 0/90 boundary was observed. Further,
the results obtained were compared with those determined by the Classical lamination theory (CLT) and it was
concluded that HDM can be applied for measurement of stress distribution across the entire thickness of the
specimen with good accuracy.
lassical lamination theory is based upon the concepts of linear ~nieotropic
elasticity. Beceuse of the stress and deformation hypotheses tt ~t are an insepar- able p~rt ofcl~ssicalamination
theory, a more correct name would be classical thin lamination theory , or even classical laminated plate theory .
Strictly speaking, this theory is valid only for a solid homogeneous continuum, which is subjected to homogeneous
boundary conditions. The approach and background.
Therml/moisture effect of CTE on laminated plates
Factors affecting the CTE of a composite were discussed previously. Other factors include chemical and physical
changes in the resin due to aging and sola physical radiation. It is significant to note that even during a CTE test
under lab- oratory conditions, the test specimen is not free from some of these effects, sug- gesting that the measured
thermal strains in reality are the sum of strains caused by di.fferent effects.
can refer to [21,22]. So, the aim of the present work is to extend the previously developed method to take into
account the thermo-mechanical coupling for laminated composite and sandwich beams. In this way, the temperature
and the displacements are written under the form of a sum of products of unidimensional polynomials of x and z.
For the mechanical unknowns, a piecewise fourth-order Lagrange polynomial of z is chosen as it is particularly
suitable for such laminated structures (see [19]). The temperature unknowns are also interpolated with the same
order expansion. As far as the variation with respect to the axial coordinate is concerned, a 1D three-node FE is
employed. Using this method, each unknown function of x is classically approximated using one degree of freedom
(dof) per node of the mesh and the LW unknown functions of z are global for the whole beam. Finally, the deduced
thermal and mechanical non-linear problems imply the resolution of two linear problems alternatively. This process
yields to two 1D problems in which the number of unknowns is much smaller than a classical Layerwise approach.
What are all the steps to calculate Von mises stress theory .
Step 1: Calculate the three principal stresses (σ1,σ2, and σ3) using principal stress equations or Mohr’s circle
method.,Step 2: Find out the Von Mises Stress (σv) following the equations mentioned above.Step 3: Determine the
value of the allowable stress (σy/N) of the material. σy is the Yield Strength and N is the factor of safety.Step 4:
Compare the value calculated in step 2 with the allowable value found in step 3. If the Value at step 2 is less than the
allowable value calculated at step 3, then the design is safe as per the Von Mises stress theory.
A rectangular plate of the sides ‘a’ and ‘b’ and total thickness ‘h’ as
shown in Figure 1 is considered.The plate consists of N number of
homogenous layers. All the layers are perfectly bounded together
and made up of linearly elastic and orthotropic material. The plate
occupies the region 0 ≤ x ≤ a, 0 ≤
≤ b h/2 ≤ ≤ h/2 in Cartesian coordinate s stem
Buckling terminology The topic of buckling is still unclear because the keywords of “stiffness”, “long” and
“slender” have not been quantified. Most of those concepts were developed historically from 1D studies. You need
to understand those terms even though finite element analysis lets you conduct buckling studies in 1D, 2D, and
3D. For a material, stiffness refers to either its elastic modulus, E, or to its shear modulus, G = E / (2 + 2 v) where v
is Poisson’s ratio.
In the past, numerous works have been carried out on the static properties of fiber reinforced polymer composites
(FRPCs). However, the dynamic properties of polymer matrix composites (PMCs) have not been fully investigated
for structural engineering applications. Determination of natural frequencies and mode shapes of p olymer matrix
composites has gained a lot of importance recently as it can help predict the possibility of resonance so that
measures can be taken to avoid it or reduce the high amplitudes associated with it. The high resonant amplitudes are
a major cause for a number of failures occurring in the aerospace industry.
In the distortion energy theory, yielding occurs when the von Mises stress σ ′ is reached, or exceeded, by a state
of stress in the machine component .. A distortion-energy theory was prompted from the observation that ductile
materials stressed hydrostatically exhibited yield strengths greatly in excess of the values given by simple tension
tests. It was then postulated that yielding was related somehow to the angular distortion of the stressed element,
rather than that yielding was a simple tensile or compressive phenomenon. Nowadays, the distortion-energy theory
for ductile materials states that yielding occurs when the distortion strain energy per unit volume reaches or exceeds
the distortion strain energy per unit volume for yield in simple tension or compression of the same mate
b Buckling Analysis of Laminated Composite Plates
In this section, an analytical solution for the buckling analysis of plate is developed using Navier solution
technique. The governing equations of the plate in case of static buckling are obtained by discarding transverse
load (q) and inertia terms (I0, I1, I2, I3) from Eqs. (13) – (16). The in-plane compressive ( 00 0 and N ,N N xx yy
xy ) forces now represents loads instead of reaction forces, as there is no transverse load. The values of in-plane
compressive forces are taken as 0 N kN xx 1 0 , 0 N kN yy 2 0 and
0 0 Nxy . The governing equations for static buckling are as follows:
Bending Analysis of Laminated Composite Plates Following the Navier solution technique, the governing equations
of the simply supported laminated composite plates in case of bending analysis are obtained by discarding in-plane
compressive loads ( 000 N ,N ,N xx yy xy ) and
In this work, the natural frequencies and mode shapes of a number of cantilever glass fiber reinforced polymer
composites (GFRPCs) and carbon fiber reinforced polymer composites (CFRPCs) are numerically obtained using
the commercial finite element analysis software (ANSYS). The laminates under study include 8 ply cantilevered
plates having a plate aspect ratio of 2 and fiber volume fractions of 0.3, 0.4, 0.5, and 0.6. The finite element analysis
procedure is described. The natural frequencies and mode shapes calculated using ANSYS are first validated with
the results obtained from previous literature. The agreement between the two results is found to be excellent. The
effect of change in the matrix material, hybridization, and laminate stacking sequence on the natural frequencies and
mode shapes are also investigated. It is found that hybridization and orientation of the outermost layer has more
significant influence on the natural frequencies of the laminated composite plates compared to fiber volume fraction
and change in the matrix material.
The natural frequencies of CFRP composite plates are found to be higher than that of GFRP composites. S2
fiberglass/epoxy plates have higher natural frequencies in comparison to E-glass/epoxy plates. However, T300
carbon/epoxy plates and T800 carbon/epoxy plates have almost similar natural frequencies. • As the fiber volume
fraction is increased, the natural frequencies corresponding to a given mode also increases. This increase is marginal
for the first three modes but comparatively higher for higher modes. • The natural frequencies of laminates with
epoxy as the matrix material and vinyl ester as the matrix material are almost the same in case of GFRP composite
plates. However, CFRP composite plates with vinyl ester as matrix material show higher natural frequencies in
comparison to CFRP composite plates with Epoxy as the matrix material. • Hybrid E-glass T300 carbon/Epoxy
plates have natural frequencies higher than that of E-glass/epoxy plates, but lower than that of T300 carbon/epoxy
plates. Moreover, [0°c /45°g /-45°g /90°c ]s hybrid plates have higher natural frequencies compared to [0°g /45°c /-
45°c /90°g ]S hybrid plates.
Residual stresses can add up to the applied load and may critically degrade the strength and integrity of the
composite structure. These stresses are particularly tricky as they satisfy equilibrium and offer no external indication
of their presence.
which affect the laminate CTE : fiber and void volumes, layup angle, fabric skewness, stacking sequence, thermal
cycling, temperature dependence, moisture effects, and material viscoelasticity
Moisture Effects ;The dimensional stability nf composites is highly affected by exposure to com- plex hygrothermal
histories. Moisture causes swelling and plasticization of the
resin system. The swelling phenomenon alters internal stresses, thus causing a dimensional change in the laminate.
When structures, machines, and engineering details are in service, they are usually subjected to surface forces or
body forces (inertial, gravitational, or electromagnetic) that cause combined stresses in their elements. To
completely define an element, it is necessary to specify the components of the stress tensor represented by the array:
The thermo-mechanical coupling for laminated composite and sandwich beams. In this way, the temperature and the
displacements are written under the form of a sum of products of unidimensional polynomials of x and z. For the
mechanical unknowns, a piecewise fourth-order Lagrange polynomial of z is chosen as it is particularly suitable for
such laminated structures (see [19]). The temperature unknowns are also interpolated with the same order expansion.
As far as the variation with respect to the axial coordinate is concerned, a 1D three-node FE is employed. Using this
method, each unknown function of x is classically approximated using one degree of freedom (dof) per node of the
mesh and the LW unknown functions of z are global .
Residual stress measurements by applying HDM using strain gauges
Sicot et al. [18] investigated the effects of cooling conditions on the development of residual stresses in [08] and
[02/902]s Carbon/epoxy [CE] composite. The models proposed by So the and Lake [15] were used to determine the
residual stresses by employing the incremental hole drilling method (IHDM). Calibration coefficients required for
residual stress determination were obtained using finite element analysis (FEA). The stresses in [08] CE composite
laminas increased with the depth of increment and a stress increase across the 0/90 boundary was observed. Further,
the results obtained were compared with those determined by the Classical lamination theory (CLT) and it was
concluded that HDM can be applied for measurement of stress distribution across the entire thickness of the
specimen with good accuracy.
Classical lamination theory is based upon the concepts of linear ~nieotropic
elasticity. Beceuse of the stress and deformation hypotheses tt ~t are an insepar- able p~rt ofcl~ssicalamination
theory, a more correct name would be classical thin lamination theory , or even classical laminated plate theory .
Strictly speaking, this theory is valid only for a solid homogeneous continuum, which is subjected to homogeneous
boundary conditions. The approach and background.
Therml/moisture effect of CTE on laminated plates
Factors affecting the CTE of a composite were discussed previously. Other factors include chemical and physical
changes in the resin due to aging and sola physical radiation. It is significant to note that even during a CTE test
under lab- oratory conditions, the test specimen is not free from some of these effects, sug- gesting that the measured
thermal strains in reality are the sum of strains caused by di.fferent effects.
can refer to [21,22]. So, the aim of the present work is to extend the previously developed method to take into
account the thermo-mechanical coupling for laminated composite and sandwich beams. In this way, the temperature
and the displacements are written under the form of a sum of products of unidimensional polynomials of x and z.
For the mechanical unknowns, a piecewise fourth-order Lagrange polynomial of z is chosen as it is particularly
suitable for such laminated structures (see [19]). The temperature unknowns are also interpolated with the same
order expansion. As far as the variation with respect to the axial coordinate is concerned, a 1D three-node FE is
employed. Using this method, each unknown function of x is classically approximated using one degree of freedom
(dof) per node of the mesh and the LW unknown functions of z are global for the whole beam. Finally, the deduced
thermal and mechanical non-linear problems imply the resolution of two linear problems alternatively. This process
yields to two 1D problems in which the number of unknowns is much smaller than a classical Layerwise approach.
