**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.

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__MEMBRANE STRESS CHECK__

__Check for Data1:__

Sx = 0.81 N/mm^{2}

Sy = 9.28 N/mm^{2}

Txy = 6.4 N/mm^{2}

SP1 = (0.81+9.28)/2 + Ѵ ((0.81-9.28)^{2}/4 + 6.4^{2}) = 12.72 N/mm^{2}

SP2 = (0.81+9.28)/2 – Ѵ ((0.81-9.28)^{2}/4 + 6.4^{2}) = -2.63 N/mm^{2}

*Hence, SP (Maximum Combined SP in SACS output) = 12.72 N/mm ^{2}*

__Check for Data2:__

Sx = -0.81 N/mm^{2}

Sy = -9.28 N/mm^{2}

Txy = -6.4 N/mm^{2}

SP1 = (-0.81-9.28)/2 + Ѵ ((-0.81+9.28)^{2})/4 + -6.4^{2}) = 2.63 N/mm^{2}

SP2 = (-0.81-9.28)/2 – Ѵ ((-0.81+9.28)^{2})/4 + -6.4^{2}) = -12.72 N/mm^{2}

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

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__BENDING STRESS CHECK:__

__Check for Data1:__

Sx = -1. 98 N/mm^{2}

Sy = -7.4 N/mm^{2}

Txy = -2.08 N/mm^{2}

SP1 = (-1.98-7.4)/2 + Ѵ ((-1.98+7.4)^{2})/4 + -2.08^{2}) = -1.23 N/mm^{2}

SP2 = (-1.98-7.4)/2 – Ѵ ((-1.98+7.4)^{2})/4 + -2.08^{2}) = -8.11 N/mm^{2}

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

__Check for Data2:__

Sx = 1. 98 N/mm^{2}

Sy = 7.4 N/mm^{2}

Txy = 2.08 N/mm^{2}

SP1 = (1.98+7.4)/2 + Ѵ ((1.98-7.4)^{2})/4 + 2.08^{2}) = 8.11 N/mm^{2}

SP2 = (1.98+7.4)/2 - Ѵ ((1.98-7.4)^{2})/4 + 2.08^{2}) = 1.23 N/mm^{2}

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

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__COMBINED PRINCIPAL STRESS AND MAXIMUM SHEAR STRESS:__

__Check for Data1:__

Sx membrane = 0.81 N/mm^{2}; Sx bending = -1.98 N/mm^{2}

Sy membrane = 9.28 N/mm^{2}; Sy bending = -7.4 N/mm^{2}

Txy membrane = 6.4 N/mm^{2}; Txy bending = -2.08 N/mm^{2}

__Case1:__

Sx1 = Sx membrane + Sx Bending = 0.81-1.98 = -1.17 N/mm^{2}

Sy1 = Sy membrane + Sy bending = 9.28-7.4 = 1.88 N/mm^{2}

Txy1 = Txy membrane + Txy bending = 6.4-2.08 = 4.32 N/mm^{2}

SP1_case1 = 4.94 N/mm^{2 }(by Equation 1)

SP2_case1 = -4.23 N/mm^{2 }(by Equation 2)

Tmax_case1 = (4.94+4.23)/2 = 4.58 N/mm^{2 }(by Equation 2)

__Case2:__

Sx2 = Sx membrane - Sx Bending = 0.81+1.98 = 2.79 N/mm^{2}

Sy2 = Sy membrane - Sy bending = 9.28+7.4 = 16.68 N/mm^{2}

Txy2 = Txy membrane - Txy bending = 6.4+2.08 = 8.48 N/mm^{2}

SP1_case2 = **20.7** N/mm^{2 }(by Equation 1)

SP2_case2 = -1.23 N/mm^{2 }(by Equation 2)

Tmax_case2 = (20.7+1.23)/2 = **10.96** N/mm^{2 }(by Equation 2)

*Hence, maximum combined principal stress SP, reported in SACS output = 20.7 N/mm ^{2}*

*and Tmax reported in SACS output = 10.96 N/mm ^{2}*

__Check for Data2:__

Sx membrane = -0.81 N/mm^{2}; Sx bending = 1.98 N/mm^{2}

Sy membrane = -9.28 N/mm^{2}; Sy bending = 7.4 N/mm^{2}

Txy membrane = -6.4 N/mm^{2}; Txy bending = 2.08 N/mm^{2}

__Case1:__

Sx1 = Sx membrane + Sx Bending = -0.81+1.98 = 1.17 N/mm^{2}

Sy1 = Sy membrane + Sy bending = -9.28+7.4 = -1.88 N/mm^{2}

Txy1 = Txy membrane + Txy bending = -6.4+2.08 = -4.32 N/mm^{2}

SP1_case1 = 4.23 N/mm^{2 }(by Equation 1)

SP2_case1 = -4.94 N/mm^{2 }(by Equation 2)

Tmax_case1 = (4.23+4.94)/2 = 4.58 N/mm^{2 }(by Equation 2)

__Case2:__

Sx2 = Sx membrane - Sx Bending = -0.81-1.98 = -2.79 N/mm^{2}

Sy2 = Sy membrane - Sy bending = -9.28-7.4 = -16.68 N/mm^{2}

Txy2 = Txy membrane - Txy bending = -6.4-2.08 = -8.48 N/mm^{2}

SP1_case2 = 1.23 N/mm^{2 }(by Equation 1)

SP2_case2 = **-20.7** N/mm^{2 }(by Equation 2)

Tmax_case2 = (1.23+20.7)/2 = **10.96** N/mm^{2 }(by Equation 2)

*Hence, maximum combined principal stress SP, reported in SACS output = -20.7 N/mm ^{2}*

*and Tmax reported in SACS output = 10.96 N/mm ^{2}*

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The snap below provides the summary of the calculations.^{}

__VON MISES STRESS:__

__Check for Data1:__

SP1_case1 = 4.94 N/mm^{2 }(See Section 3 of this Calculation Note)

SP2_case1 = -4.23 N/mm^{2 }(See Section 3 of this Calculation Note)

SP1_case2 = 20.7 N/mm^{2 }(See Section 3 of this Calculation Note)

SP2_case2 = -1.23 N/mm^{2 }(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/mm^{2}

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

*Hence, maximum von Mises stress reported in SACS output = 21.33 N/mm ^{2}*

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__Check for Data2:__

SP1_case1 = 4.23 N/mm^{2 }(See Section 3 of this Calculation Note)

SP2_case1 = -4.94 N/mm^{2} (See Section 3 of this Calculation Note)

SP1_case2 = 1.23 N/mm^{2 }(See Section 3 of this Calculation Note)

SP2_case2 = -20.7 N/mm^{2 }(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/mm^{2}

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

*Hence, maximum von Mises stress reported in SACS output = 21.33 N/mm ^{2}*

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