Combined Plate Stress calculation in SACS:

To illustrate the calculation method, let us take a sample SACS Plate Stress Result data as below

For conventions of Membrane & Bending stresses in Plate element, refer SACS POST manual.

Computations of principal stresses are given by the following equations.

     

 

The maximum stress combinations used in SACS program is given as below (see SACS POST Manual)

To evaluate the plate combined stresses here, 2 sets of data is considered which are obtained from a sample SACS analysis. The first line of the represents Data 1 and the second line represents Data2 respectively of the above SACS output.

 

1. MEMBRANE STRESS CHECK

Check for Data1:

Sx = 0.81 N/mm²

Sy = 9.28 N/mm²

Txy = 6.4 N/mm²

SP1 = (0.81+9.28)/2 + Ѵ ((0.81-9.28)²/4 + 6.4²) = 12.72 N/mm²

SP2 = (0.81+9.28)/2 – Ѵ ((0.81-9.28)²/4 + 6.4²) = -2.63 N/mm²

Hence, SP (Maximum Combined SP in SACS output) = 12.72 N/mm²

Check for Data2:

Sx = -0.81 N/mm²

Sy = -9.28 N/mm²

Txy = -6.4 N/mm²

SP1 = (-0.81-9.28)/2 + Ѵ ((-0.81+9.28)²)/4 + -6.4²) = 2.63 N/mm²

SP2 = (-0.81-9.28)/2 – Ѵ ((-0.81+9.28)²)/4 + -6.4²) = -12.72 N/mm²

Hence, SP (Maximum Combined SP in SACS output) = -12.72 N/mm2

 

2. BENDING STRESS CHECK:

Check for Data1:

Sx = -1. 98 N/mm²

Sy = -7.4 N/mm²

Txy = -2.08 N/mm²

SP1 = (-1.98-7.4)/2 + Ѵ ((-1.98+7.4)²)/4 + -2.08²) = -1.23 N/mm²

SP2 = (-1.98-7.4)/2 – Ѵ ((-1.98+7.4)²)/4 + -2.08²) = -8.11 N/mm²

Hence, SP (Maximum Combined SP in SACS output) = -8.11 N/mm2

Check for Data2:

Sx = 1. 98 N/mm²

Sy = 7.4 N/mm²

Txy = 2.08 N/mm²

SP1 = (1.98+7.4)/2 + Ѵ ((1.98-7.4)²)/4 + 2.08²) = 8.11 N/mm²

SP2 = (1.98+7.4)/2 - Ѵ ((1.98-7.4)²)/4 + 2.08²) = 1.23 N/mm²

Hence, SP (Maximum Combined SP in SACS output) = 8.11 N/mm2

 

3. COMBINED PRINCIPAL STRESS AND MAXIMUM SHEAR STRESS:

Check for Data1:

Sx membrane = 0.81 N/mm²; Sx bending = -1.98 N/mm²

Sy membrane = 9.28 N/mm²; Sy bending = -7.4 N/mm²

Txy membrane = 6.4 N/mm²; Txy bending = -2.08 N/mm²

Case1:

Sx1 = Sx membrane + Sx Bending = 0.81-1.98 = -1.17 N/mm²

Sy1 = Sy membrane + Sy bending = 9.28-7.4 = 1.88 N/mm²

Txy1 = Txy membrane + Txy bending = 6.4-2.08 = 4.32 N/mm²

SP1_case1 = 4.94 N/mm² (by Equation 1)

SP2_case1 = -4.23 N/mm² (by Equation 2)

Tmax_case1 = (4.94+4.23)/2 = 4.58 N/mm² (by Equation 2)

 

Case2:

Sx2 = Sx membrane - Sx Bending = 0.81+1.98 = 2.79 N/mm²

Sy2 = Sy membrane - Sy bending = 9.28+7.4 = 16.68 N/mm²

Txy2 = Txy membrane - Txy bending = 6.4+2.08 = 8.48 N/mm²

SP1_case2 = 20.7 N/mm² (by Equation 1)

SP2_case2 = -1.23 N/mm² (by Equation 2)

Tmax_case2 = (20.7+1.23)/2 = 10.96 N/mm² (by Equation 2)

Hence, maximum combined principal stress SP, reported in SACS output = 20.7 N/mm²

and Tmax reported in SACS output = 10.96 N/mm²

 

Check for Data2:

Sx membrane = -0.81 N/mm²; Sx bending = 1.98 N/mm²

Sy membrane = -9.28 N/mm²; Sy bending = 7.4 N/mm²

Txy membrane = -6.4 N/mm²; Txy bending = 2.08 N/mm²

Case1:

Sx1 = Sx membrane + Sx Bending = -0.81+1.98 = 1.17 N/mm²

Sy1 = Sy membrane + Sy bending = -9.28+7.4 = -1.88 N/mm²

Txy1 = Txy membrane + Txy bending = -6.4+2.08 = -4.32 N/mm²

SP1_case1 = 4.23 N/mm² (by Equation 1)

SP2_case1 = -4.94 N/mm² (by Equation 2)

Tmax_case1 = (4.23+4.94)/2 = 4.58 N/mm² (by Equation 2)

 

Case2:

Sx2 = Sx membrane - Sx Bending = -0.81-1.98 = -2.79 N/mm²

Sy2 = Sy membrane - Sy bending = -9.28-7.4 = -16.68 N/mm²

Txy2 = Txy membrane - Txy bending = -6.4-2.08 = -8.48 N/mm²

SP1_case2 = 1.23 N/mm² (by Equation 1)

SP2_case2 = -20.7 N/mm² (by Equation 2)

Tmax_case2 = (1.23+20.7)/2 = 10.96 N/mm² (by Equation 2)

Hence, maximum combined principal stress SP, reported in SACS output = -20.7 N/mm²

and Tmax reported in SACS output = 10.96 N/mm²

 

The snap below provides the summary of the calculations.

4. VON MISES STRESS:

Check for Data1:

SP1_case1 = 4.94 N/mm² (See Section 3 of this Calculation Note)

SP2_case1 = -4.23 N/mm² (See Section 3 of this Calculation Note)

SP1_case2 = 20.7 N/mm² (See Section 3 of this Calculation Note)

SP2_case2 = -1.23 N/mm² (See Section 3 of this Calculation Note)

 

Hence, von Mises stress is given by equation 4 as

VM_case1 = [4.94^2 + -4.23^2 – (4.94*-4.23)]^0.5 = 7.94 N/mm²

VM_case2 = [20.7^2 + -1.23^2 – (20.7*-1.23)]^0.5 = 21.33 N/mm²

Hence, maximum von Mises stress reported in SACS output = 21.33 N/mm²

 

Check for Data2:

SP1_case1 = 4.23 N/mm² (See Section 3 of this Calculation Note)

SP2_case1 = -4.94 N/mm² (See Section 3 of this Calculation Note)

SP1_case2 = 1.23 N/mm² (See Section 3 of this Calculation Note)

SP2_case2 = -20.7 N/mm² (See Section 3 of this Calculation Note)

 

Hence, von Mises stress is given by equation 4 as

VM_case1 = [4.23^2 + -4.94^2 – (4.23*-4.94)]^0.5 = 7.94 N/mm²

VM_case2 = [1.23^2 + -20.7^2 – (1.23*-20.7)]^0.5 = 21.33 N/mm²

Hence, maximum von Mises stress reported in SACS output = 21.33 N/mm²