In most software, the stress is already calculated at the gauss points as noted by Alejandro.  In general, stress is not directly solved for by the finite element method.  Displacements are calculated at the nodes.  All other displacements are interpolated from the shape functions.  The stress (and resulting strain) is found by taking the derivative terms from the shape functions.  Hence stress and strain are never calculated directly, but are interpolated from these derivative terms and the nodal displacements.  It turns out, the stress happens to be most accurately interpolated at the Gauss points in the isoparametric formulation, and most commercial software will output the stress at these locations in a straightforward manner.  Please note that if stress at a particular node is desired, it will be different when calculated from each element attached to that node.  Care should be taken when combining the stress from multiple elements at a node.

Dear Bhushan

In the FEM modeling, it is possible only  to calculate the stresses in the nodes. the Gauss points are introduced for numerical integration and we can not obtain the responses in these points. This procedure is possible in the BEM approach using internal points.

best regards

Panji

Dear Awani,

If Gauss points are being used to compute the stiffness matrix (most FE software do it since exact integration is only possible in few cases), then the stresses are already being calculated at the Gauss points. So, when plotting the stresses these FE programs what really do by default is to plot the stresses at the Gauss points. If you want the stresses at the nodes, what you can do is to make a smoothing or averaging around the nodes (most FE software have an option to do this). In fact, the computation of stresses directly at the nodes is in many situations not possible in FE since most of the FE basis functions are C0 functions and so they are not differentiable at the nodes locations.

Best.

I agree with the response of Alejandro.

In most software, the stress is already calculated at the gauss points as noted by Alejandro.  In general, stress is not directly solved for by the finite element method.  Displacements are calculated at the nodes.  All other displacements are interpolated from the shape functions.  The stress (and resulting strain) is found by taking the derivative terms from the shape functions.  Hence stress and strain are never calculated directly, but are interpolated from these derivative terms and the nodal displacements.  It turns out, the stress happens to be most accurately interpolated at the Gauss points in the isoparametric formulation, and most commercial software will output the stress at these locations in a straightforward manner.  Please note that if stress at a particular node is desired, it will be different when calculated from each element attached to that node.  Care should be taken when combining the stress from multiple elements at a node.

Dear Awani, In FEM, nodes are connected to more than one elements (in general) and have no volume. So basically, we can only calculate the displacements/forces on the nodes. We cannot define a strain or stress on the node. The strain and stress can only be defined defined on the element (I.e. Gauss point or integration point). When you see nodal stresses and strains in a FEM analysis then that is actually an averaged (smoothed) value of the stresses/strains obtained in the neighboring integration points of the elements to which the node is connected.

Dear Bhushan,

In FEM analysis generally the displacement is calculated at nodes in some commercial software. The stresses and starins are calculated  at the gauss point. so in order to find out stresses at node, generally interpolation is used in most of FEM softwares. The different options are available in such software.  

U can multiply the value at the quadrature point by

[1+sqrt(3)/2 -1/2 1-sqrt(3)/2 -1/2;

-1/2 1-sqrt(3)/2 -1/2 1+sqrt(3)/2 ;

1-sqrt(3)/2 -1/2 1+sqrt(3)/2 -1/2;

-1/2 1+sqrt(3)/2 -1/2 1-sqrt(3)/2 ] ;

It will extrapolate the value to node.

Dear Awani,

Most of the numerical integration will be performed using Gauss point and this method is used to calculate stress as well as strain at the Gauss point. the finite element equation is developed based on displacement model and displacement within the volume element is calculated based on shape function. so, it is possible to calculated the stress at Gauss point with the use of known displacement at the Gauss point.. ok all the best in your work..

Dear all,

Could you help me to answer the question "How to calculate the stress at the centroid of an Hexagonal element from the results of Stresses at 8 Gaussian points?

Thank you very much.

@T-D Hoang, I assume you wanted to say hexahedron. If so, you could define another hexahedron with the coordinates of the Gauss points. These Gauss points are now the nodes of the new hexahedron. Then, the stresses at the Gauss points become nodal stressess of the new hexahedron. Now, to compute the stresses at the centroid, you can use the shape functions of the new hexahedron along with its nodal stresses to interpolate the stresses at the centroid of the hexahedron.

Alejandro Ortiz-Bernardin . Thank you for your answer. Yes, it is a hexahedron 8 nodes.

So the coordinate of the centroid is x(0, 0, 0). If I used the interpolation with the shape functions as below code manner

rvalue=dsqrt(3.d0);

p=rvalue*x(1)=0;

q=rvalue*x(2,)=0;

r=rvalue*x(3)=0;

Vector(1)=.125*(1.d0-p)*(1.d0-q)*(1.d0-r);

Vector(2)=.125*(1.d0+p)*(1.d0-q)*(1.d0-r);

Vector(3)=.125*(1.d0+p)*(1.d0+q)*(1.d0-r);

Vector(4)=.125*(1.d0-p)*(1.d0+q)*(1.d0-r);

Vector(5)=.125*(1.d0-p)*(1.d0-q)*(1.d0+r);

Vector(6)=.125*(1.d0+p)*(1.d0-q)*(1.d0+r);

Vector(7)=.125*(1.d0+p)*(1.d0+q)*(1.d0+r);

Vector(8)=.125*(1.d0-p)*(1.d0+q)*(1.d0+r);

do k=1,6

do igauss=1,8 EStresses_Centroid(k)=Vector(igauss)*EStresses_Gausspoint(k,igauss)

enddo

enddo

Hence,

EStresses_Centroid(k) = 1/8*(EStresses_Gausspoint(k,igauss)) = Average(EStresses_Gausspoint(k,igauss)

So, the stress at the centroid is the average of all stresses at 8 Gauss points. Is it correct?

I look forward to getting your kind reply.

Thank you.

@T-D Hoang, yes, it is correct.

Alejandro Ortiz-Bernardin Thank you very much!