Combined Loading Effects in Engineering

In engineering design, engineers are generally dealing with the combined effects of loading on different mechanical parts or structures. Because of these different combined loading conditions, there can be different kinds of combined stresses that can occur on a design point. In this case, an engineer must understand the combined loading conditions and must apply design parameters to reach a safe design. 

One can understand the combined loading effects on mechanical elements if one takes a look at this circular shaft example. You can use these formulations in different engineering problems. 

Combined Loading Effects 

Combined loading on a circular shaft.
Combined loading on a circular shaft(Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.627).

Think about a solid shaft that undergoes different combined loading conditions such as twisting moment and bending force. A singular force at the end of the shaft may produce these two effects on the shaft. Because of these forces, shear force, twisting moment, and bending moment occur on the shaft cross-section.

According to your design or structure, you need to define the most critical point where failure would occur because of the different combined loading conditions. And you need to make your safe design according to this point. If the combined stresses do not exceed the safety margin, your structure will be safe.

Combined effects of cross-section.
Combined effects of cross-section (Image Source:D. K. Singh – Strength of Materials-Springer, 2020, pg.627).

But first of all, you can calculate the different combined loading conditions one by one on that point to find the total combined stress because of these singular loads.

Calculation of Twisting Moment in Combined Loads

If we take a look at the effect of the twisting moment which can be produced with the action of combined loads, torsional shear stress will occur on the cross-section of the shaft. So, maximum torsional will occur on the surface. 

If we apply the torsion formula; 

Torsional load formula.

T is the twisting moment that is applied on the shaft. ‘d’ is the diameter of the shaft. Hence, the calculation of the torsional effect on the shaft is very simple as that. You can use the calculator below to calculate it easily. 

Torsional Stress Calculator













The use of the calculator above is very simple. You just need to enter the required values stated above. Then click on the ‘Calculate!’ button to see the result. If you want to make another calculation, click on the ‘Reset’ button. 

Calculation of the Bending Moment in Combined Loads

Bending moment formula.

Because of the bending moment acting on the shaft, there will be tensile effect would occur. And also, if we take a look at the calculation of this tensile stress; 

In this formula, ‘M’ is the value of the bending moment, and ‘d’ is the diameter of the shaft. You can also use the calculator below.

Bending Tensile Stress













Calculation of Bending Force

Bending force formula.

Again, the bending force on the shaft produces a shear force which must be included in the calculations. And also at the outermost surface, the effect of the shear force is maximum. Furthermore let’s take a look at the calculation of the shear force effect on a shaft; 

Again here, ‘V’ is the vertical bending force and the ‘d’ is the diameter of the shaft. You can use the calculator below to calculate it. 

Shear Stress













Calculation of the Principal Stresses

To make the safety calculations, you need to calculate the principal stresses because of these combined stress effects. So, you can use this link to calculate the principal stresses for your system. And also, after the calculation of your principal stresses, you can use different stress theories to calculate the safety of your system. 

Combined Torsion and Axial Combined Loads

Circulat shaft.
Representation of a circular shaft(Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.638).

A circular shaft will be the best representative of mechanical elements to show the combined loading effects on it. Furthermore, you can apply to your systems the general principles that are applied here on the circular shaft. 

Think about a circular shaft undergoing the effects of torsion and axial combined loads. There is a twisting moment(T) and axial load(P). 

Shear stress(τ) is induced on the shaft because of the twisting moment and normal stress(σ) is induced because of the combined axial load. Let’s take a look at these values. 

Shear Stress Calculation Due to Twisting Moment

If a twisting moment is applied to a circular shaft, there will be shear stress-induced on this shaft. And the biggest shear ones occurs on the surface of the shaft. Shear stress can be calculated via this formula; 

Shear stress formula for combined loads.

In this formula, ‘D’ is the diameter that the shear one occurs. If you want to see the maximum shear, you need to use this formulation for circular shafts. The 0.098 comes from the polar moment of inertia of the circular cross-section. You need to use the polar moment of inertia formulations for different cross-sections.

You can use the calculator below to calculate the shear due to the twisting moment on a structural element. The use of this calculator is very simple. You just need to enter the required values inside the brackets, then click on the ‘Calculate!’ button. If you want to make another calculation, just click on the ‘Reset’ button then enter the new values.

Torsional Stress Calculator













Normal Stress Calculation Due to the Axial Load

As we stated above, combined axial load causes normal stress perpendicular to the cross-section of the shaft. So, there is no maximum or minimum point of this normal. Normal stress is the same for all sides of the cross-section. The normal stress can be calculated via this formula; 

In this formula, ‘F’ is the normal force acting axially to the shaft and the ‘D’ is the diameter of the shaft. 

You can use the calculator below to calculate combined normal stress due to the normal force acting along the axis of the structural elements. 

Normal stress calculation.

In this formula, ‘F’ is the normal force acting axially to the shaft and the ‘D’ is the diameter of the shaft. 

You can use the calculator below to calculate normal stress due to the normal force acting along the axis of the structural elements. 

Normal Stress Calculator













Calculation of Combined Effects Loads

The combined effects must be obtained from these two types of combined stresses. So, the acting directions of these combined stresses must be well defined by you to obtain a safe design. You need to show the stresses on stress elements and find out the principal stresses via formulations or Mohr’s circle method. 

Once you find the maximum principal stresses on a critical point, you need to compare this stress result to the combined safety value that material remains durable. This generally yield the strength of the material of the part which is divided by the safety factor. 

Combined Torsion, Bending, and Direct Thrust Loads

Circular element.
Image Source: D. K. Singh – Strength of Materials-Springer, 2020, pg.641.

In the case of marine propellers, torsion takes place because of the rotation of the shaft due to the power transmission. Bending takes place because of the self-weight of the shaft which is fixed with bearings. Direct thrust force is the result of the rotation of the propeller at the edge of the shaft, which applies an axial force along the shaft. 

Calculation of Twisting Moment Effect

Because of the rotation, a twisting moment occurs on the shaft. Also, the effect of the twisting moment is the shear which is the maximum at the surface of the shaft. This equation is used to calculate the effect of the twisting moment; 

Twisting moment formula for combined loadings.

In here, ‘T’ is the twisting moment, ‘d’ is the diameter of the shaft.

You can use the calculator below to calculate shear due to the twisting moment effect to make more fast calculations.

Torsional Stress Calculator













The use of this calculator is very simple. You just need to enter the required values inside the brackets and then click on the ‘Calculate’ button to see the result. If you want to make another calculation, just click on the ‘Reset’ button and then re-enter the new values. 

Calculation of Bending Moment Effect

The bending moment occurs because of the own weight of the shaft. The bending moment must be calculated according to the placement of the bearings. The formulation to calculate the tensile or compressive bending stress is like this; 

Bending stress formula.

In this equation, ‘M’ is the magnitude of the bending moment.

Use the calculator below to calculate the tensile or compressive combined bending stress due to the bending moment.

Tensile Or Compressive Stress Calculator













Calculation of Vertical Shear Force Effect

Shear force is not produced at the top-side surface of the shaft. The maximum shear is occurred at the side surface of the shaft because of the shear force effect. This combined shear stress is calculated via this formulation; 

Shear stress for combined loads.

In this formulation, ‘V’ is the vertical shear force. You can use the calculator below to swiftly calculate the shear stress due to the shear force.

Calculator Of Shear Stress Due To Shear Force













Calculation of Combined Direct Thrust Stress Effect

Compressive stress occurred on the shaft because of the thrust force produced by the propeller. The calculation is made with this equation; 

Thrust force.

In this equation, ‘t’ is the thrust force.

Use the calculator below to calculate the normal compressive stress due to the thrust force.

Calculator Of Normal Stress Due To Thrust Force













Find The Critical Point for Combined Load

In terms of the direct stress effects, thrust and combined bending stresses are superimposed. Furthermore, the critical point is where the maximum direct combined stress occurs on the shaft. It is generally the top surface of the shaft. The effect of shear force is the same for all the points on surfaces. 

After finding out the critical point and the stresses on that critical point, you can calculate the maximum principal stresses on that point.

Uniaxial Tension Combined With Torsional Shear Stress?

Σ1 Calculation









Σ2 Calculation









Combined torsional and uniaxial tension on Mohr's circle.
Combined torsional and uniaxial tension on Mohr’s circle.(Robert L. Moth: Machine Elements In Mechanical Design).

In general strength calculations, combined loads are a general phenomenon. At the point of a material that undergoes various types, shows three combined stress types in general;

Torsional stress which is Txy,

Tensile or compression load in the X axis which is ΣX,

Tensile or compression stress in the Y axis which is ΣY,

So, in some situations, some of these stress values can be 0. What you need to do is, you just need to write 0 on principal load formulas.

Principal stress formulas.
Principal stress formulas.

As you see above, you can calculate principal stress values from these formulas by putting all the combined stress values. Also, you can find out the radius of Mohr’s circle.

Click to see all the other engineering calculators in Mechanicalland!

How to Use Uniaxial Tension Combined with Torsional Shear Principal Stress Calculator?

What you need to do is very basic. Just type ΣX, ΣY, and Txy values that you found from your engineering problem. Also, this calculator will give you the first and second principal stress values.

Conclusion

Mechanicalland does not accept any responsibility for calculations made by users in calculators. A good engineer must check calculations again and again.

You can find out much more calculators like this in Mechanicalland! Also, take a look at the other engineering calculators available in Mechanicalland!

Do not forget to leave your comments and questions below about combined loading effects. 

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