Heat transfer analysis

Steady-State or Transient Heat Transfer Analysis is devoted to predicting the temperature distribution for an object exposed to heating, radiation, convection and conduction. Thermal Stress Analysis predicts stresses, displacements due to thermal expansion or contraction.

To perform heat transfer analysis or static structural analysis the same types of elements can be used. There is no need to decrease the order of elements or to remesh the model. For three-dimensional (3D) brick elements there are 3 nodal displacements for static analysis. Nodal temperature is an additional variable. Therefore, there are 4 variables per node for the thermal stress analysis. The structural stiffness matrix does not depend on boundary conditions or nodal temperature.

Temperature change causes thermal expansion or contraction. High temperature causes thermal expansion of a solid. Uniform heating results in deformation only, not thermal stress. High gradient of temperature causes distortion of the beam. Thermal stresses are caused by temperature gradient. The smaller the distance between nodes with different temperatures, the larger the distortion and thermal stresses.

Cooling causes contraction and tensile stress s_z in most of the elements. The stress does not depend on the width of the plate. The stress at the left-hand side of the notched specimen is very small. There is a stress gradient in the net-section A - B. The stress is the largest in element A.

Oil quenching of a steel structure can be modeled by transient heat transfer module. The results are transferred to static analysis to calculate the thermal stresses. Quenching causes contraction of external surfaces. The contraction causes the tensile stress on those surfaces. The intersection of thin and thick walls causes the highest thermal stresses.

A pressure vessel with bimetal steel walls was heated till 400^oC. The coefficient of thermal expansion is larger for stainless steel. The stainless steel layer tries to expand but the titanium alloy layer does not have such a large deformation. There is a negative tangential thermal stress in the stainless steel. There is additional compressive thermal stress in the stainless steel near the layer's intersection.

Uniform heating of the rigid ring will expand it mostly in the tangential direction. There is a smaller displacement in the radial direction. Thin walled dome-shape plate will have a larger radial displacement. It causes the central points to move upwards.

A cold drop of water on a hot metal surface causes thermal cracking due to local tensile stress. The temperature distribution in the structure is used as the loading condition for a structural analysis to calculate thermal stress.

The temperature can be calculated using the differential equation governing transient heat transfer with a heat source. In this case:

k is the heat transfer coefficient;
T is temperature;
t is time;
Q(t) is the heat source.

DYNAMICS

Modal Analysis. Understanding the natural frequencies and corresponding modes of engineering structures can help improve performance and guarantee safety. Changing the external forces causes dynamic effects. For example, vibrations are generated in vehicles from motors or road conditions, in ships from waves, in airplane wings due to turbulence, etc. The maximum stresses resulting from the vibration are considered in engineering analysis. The stresses define the lifetime of a structure.

Vibration involves repetitive motion. Frequency is defined as the number of cycles in a given time period. 10 Hertz is the same as 10 cycles per second.

The dynamic magnification factor D is equal to the ratio of amplitude at a given frequency to amplitude of the static response.

The number of characteristic frequencies of the model is equal to number of DOF in the FE model. The most important results are within the first few natural frequencies. The coincidence of the external vibration with the first (and smallest) fundamental frequency results in the maximum deflection of the structure.

The first frequency is a property of structure. It does not change if the number of FE in the model increases. FEM solves the eigenvalue problem with better approximation for low frequencies. The higher the frequency, the smaller the correspondence between the FE model and the real situation. There is no need to predict all high frequencies for a structure. It is possible to identify one structure from another by the set of natural frequencies, the "finger prints" of a structure.

The most serious consequences occur when a power-driven device produces a frequency at which an attached structure naturally vibrates. This effect is called "resonance". If sufficient power is applied, the structure can be destroyed. The major purpose of the modal analysis is to avoid resonance. Ideally, the first mode must have a frequency higher than any potential driving frequencies.

In the example, the membrane of a load cell is compressed with variable external pressure. External pressure is applied to the membrane with a frequency of 1 / (time period) = 1 / 0.2 = 5 Hz. If the FEA shows that the natural frequency of the membrane is about 5 Hz resonance will take place. The load cell cannot correctly reflect the pressure values. It is recommended to use the cell with lower frequency external pressure. If this is not possible then the cell design must be changed.

An increase of the modulus of elasticity E can increase the fundamental frequency w₁. Natural frequency decreases for heavier material.

The following linear eigenvalue problem is solved to calculate the natural frequencies and associated mode shape of a finite element model. Here
[K] is the structural stiffness matrix;
[M] is the mass matrix;
w_i is the i^th natural frequency;
{D_i} is the i^th mode shape or eigenvector.

Natural frequencies and mode shapes are the results of modal analysis. They do not depend on static loading schemes. The force vector is a zero-element vector.

Predicting the effects of impacts are the most common use of transient dynamics. The second formula given shows the dependence of dynamic response on applied force. Stiffness matrix relates forces and displacements. Inertia is described by point accelerations. Damping effect is defined by the velocity of the body.

Analysis of dynamic response by FEM is shown in the example. The solutions were obtained for a time interval Dt. The smaller the time interval the smaller the errors.
There is a critical value Dt_critical above which the step-by-step integration leads to significant error. The value of Dt_critical depends on the highest frequency of the model.

In the example a static load had caused tensile stress to reach the yield strength in the low-carbon steel thin plate. The dynamic response problem of the weight dropped onto the plate from a height of 0.5 meter is solved with nonlinear dynamic analysis. There are damping vibrations. The dynamic magnification factor is much larger than 1. This means that at dynamic loading the stress exceeds the yield strength of the material and there is a residual deformation in plate that is larger than the static one.