Checking elements 11, 12/12′ (central bar, left/right bar) with MDESIGN

In the steps below, 🖱 means a left mouse click and → leads to the next action. Labels in the screenshots are those of the MDESIGN interface.

1. Analysis model and input data

Analysis model in the XYZ coordinate system of the ATD

Analysis model in the xyz coordinate system of MDESIGN

Input data (Ex.6.1)

Forces: F_X = 1500 N, F_Y = 300 N, F_Z = 700 N.

Moments: M_X11 = 75000 Nmm, M_Y11 = 585000 Nmm, M_Z11 = 90000 Nmm.

Geometric parameters: L₁₁ = 550 mm, h₁₂ = 50 mm, t₁₂ = 5 mm, h₁₁ = 60 mm, t₁₁ = 5 mm.

Material: S235, weldable general-purpose steel.

Operating constraints: temperature T = (−30…40) °C.

Forces and moments in the MDESIGN xyz coordinate system

Forces: F_x = F_Y = 300 N, F_y = −F_X = −1500 N, F_z = −F_Z = −700 N.

Moments: T_x = M_Y11 = 585000 Nmm, M_y = M_X11 = 75000 Nmm, M_z = M_Z11 = 90000 Nmm.

2. Entering the data in MDESIGN (Annex.MD.1)

2.1 Launching the application

Opening the MDESIGN shaft (extended) module

🖱

→

🖱

→

🖱

Resetting the data of the automatically loaded application

🖱 → [🖱 select the command ] → [🖱 select the option ].

Saving the application

🖱 (, ) → : [ select the folder where the application will be saved],

[ enter the file name Elements 11+12], [: ], 🖱

[the title of the main window shows: ].

Opening the application (if it was created and saved earlier)

[select the folder where the application is saved] → 🖱 .

Note. While working, save intermediate versions regularly for safety (the system may freeze), using 🖱 or F10.

2.2 Selecting the Input window and general settings

Selecting the language

[🖱 select from the list the option ].

Selecting the Input page

Notes.

– the option is selected to view the results obtained after solving;

– the option is selected to view the input data.

General settings (selected from the attached lists)

Note. The option requires entering the position of Section 2 (the value x = L₁₁/2) in order to view the calculated results there.

2.3 Entering the geometric, technological, loading and material data

Entering data in the Beam geometry table

🖱 → 🖱 () → : select from the list 🖱, 🖱 → : select from the list 🖱 ,

🖱 (data on the chosen section appear automatically in row 1) → : enter the bar length in the box , e.g. 275;

🖱 → 🖱 () → : select from the list 🖱 , 🖱 → 🖱: select from the list 🖱 , 🖱 (data on the chosen section appear automatically in row 2), : enter the bar length in the box , e.g. 550;

🖱 → 🖱 () → : select from the list 🖱, 🖱 → : select from the list 🖱 ,

🖱 (data on the chosen section appear automatically in row 3) → : enter the bar length in the box , e.g. 275.

3D graphic view of the bar

🖱 ().

Note. The model can be manipulated with: 🖱 (, plane projection, 2D; , perspective view, 3D); 🖱 () + 🖱 (to zoom in) or + (to zoom out); 🖱 + drag to pan; + drag to rotate.

Entering the support data in the Bearings table

: 🖱, [enter the x value in the box , e.g. 0] → 🖱 () → : [🖱 select the icon for a fixed (clamped) support]; 🖱, [enter the x value in the box , e.g. 1100] → 🖱 () → : [🖱 select the icon for a fixed (clamped) support].

3D graphic view of the bar

🖱 ().

Entering the axial-force loading in the Axial Forces F_ax table

: 🖱, [enter the values in the boxes (the position, e.g. 550)], (the value, with a + sign in the direction of the x axis, e.g. 300), (the radius, i.e. the distance from the X axis, e.g. 0), (the angle in the YZ plane, measured from the Y axis, e.g. 0).

3D graphic view of the bar

🖱 ().

Entering the radial-force loading in the Radial Forces F_r table

: 🖱, [enter the values in the boxes (the axial position, e.g. 550/550)], (the value, with a − sign opposite to the axis, e.g. −1500/−700), (the angular position, e.g. 0/270)].

3D graphic view of the bar

🖱 ().

Entering the bending-moment loading in the Bending moments M_b table

: 🖱, [enter the values in the boxes (the axial position, e.g. 550/550)], (the value, with the appropriate sign, e.g. −75000/90000), (the angle in the YZ plane, measured from the Y axis, e.g. 0/90).

3D graphic view of the bar

🖱 ().

Entering the torsional-moment loading in the Torsion table

: 🖱, [enter the values in the boxes (the position), (the value of the torsional moment, e.g. 552000)] → (the length, e.g. 0)].

3D graphic view of the bar

🖱 ().

Entering the material data

[select from the list 🖱 (the MDESIGN database)].

🖱 → : [select the row with the material ()], 🖱.

Entering the working temperature ()

[enter the value in the box , e.g. 40].

Entering the options related to the semi-finished product (, )

[select from the list , 🖱 ].

Meaning

T – Temperature (maximum working temperature); Material, surface hardened; Loads mainly crosswise to the preferred direction of machining (rolling).

3. Solving the model and saving the results

🖱

or 🖱 or F10

→

Correct any inconsistencies or errors in the input data (only if warning windows appear)

→

🖱

or 🖱 or F10

→ 🖱

→

Elements 11+12.pdf

→ 🖱

4. Results obtained (Annex.MD.6.3_11,12)

Selecting the Output page

Geometry data

Meaning

l_g – total length of the beam; m – total mass of the beam; x_s – position of the centre of gravity on the X axis; I_y – moment of inertia about the y axis; I_z – moment of inertia about the z axis; W_t – torsional section modulus; W_y – section modulus about the y axis; W_z – section modulus about the z axis; S_y – first moment of area about the y axis; S_z – first moment of area about the z axis.

Loading data

Axial-force loading

Radial-force loading

Bending-moment loading

Torsional-moment loading

Material data

Meaning

R_pN – yield strength; R_mN – ultimate tensile strength.

Support reactions

Bending moment diagrams

Meaning

M_bz – bending moment development in the X–Y plane; M_by – bending moment development in the X–Z plane; M_b – resultant bending moment development.

Stress values and diagrams

Maximum normal stresses: values and diagrams

Meaning

Salient point – representative points P1, P2, P3, P3*, P4, P5, P5*, P6, P7, P7*, P8, P1* of the cross-section; σ_zd – tension-pressure (tensile/compressive) stress; σ_ηmax – bending stress due to M_ymax; σ_ζmax – bending stress due to M_zmax; σ_res – combined (resultant) normal stress; σ_bz – bending stress development in the X–Y plane; σ_by – bending stress development in the X–Z plane; σ_b – resultant bending stress development; Stress of axial force F_x – axial stress due to F_x; Stress of bending moment M_y – stress due to M_y; Stress of bending moment M_z – stress due to M_z.

Maximum shear stresses: values and diagrams

Meaning

Salient point – representative points P1, P2, P3, P3*, P4, P5, P5*, P6, P7, P7*, P8, P1* of the cross-section; τ_zmax – shear stress due to F_zmax; τ_ymax – shear stress due to F_ymax; τ_res – combined (resultant) shear stress; τ – shear stress due to the torsional moment T.

Equivalent stress and safety factor: values and diagrams

Meaning

σ_vx – equivalent stress development (resultant); σ_vxmax – maximum equivalent stress; S_F – safety against the yield point; S_Fmin – minimum safety factor against yielding.

Transverse displacements (deflections) and rotations: values and diagrams

Meaning

v – resultant transverse deflection; θ – resultant inclination (rotation); v_max – maximum resultant deflection; θ_max – maximum rotation (angle of the maximum deflection).

5. Checks

Check under combined loading

The check under combined loading aims to prevent permanent plastic deformation of the bar, based either on stresses,

σ_vxmax ≤ σ_a,

or on safety factors,

c ≥ c_a,

where σ_vxmax is the maximum equivalent stress, σ_a = σ₀₂/c_a the allowable stress against yielding (plastic deformation), σ₀₂ the yield strength and c_a the allowable (minimum) safety factor, which may be taken as c_a = 1.5…2.5; c = σ₀₂/σ_vmax is the minimum safety factor.

Applying the combined-loading check

σ_vxmax ≤ σ_a; 24.84 < 117.5 MPa (satisfied),

or

S_Fmin ≥ c_a; 9.758 > 2 (satisfied).

Notes.

– Yield strength σ₀₂ = 235 MPa (see above, the Material Data section); the allowable safety factor was taken as c_a = 2.

– If the check is not satisfied, the cross-sectional dimensions of the segments must be increased.

Check of bending deformations (stiffness)

The deformation check is carried out to prevent linear and angular displacements that are unacceptable from a functional point of view, by comparing the maximum deflections and rotations with the allowable values:

v_max ≤ v_a,

θ_max ≤ θ_a,

where v_a is the allowable deflection and θ_a the allowable rotation. The allowable deformations recommended in the literature are v_a ≤ (0.002…0.03)l_max (l_max being the maximum length of the bar) for linear deformations, and θ_a = (0.5…1.5)° for angular deformations.

Applying the deformation (stiffness) checks

Linear deformations (deflections)

v_max ≤ v_a; 0.135 ≤ 1.5 mm (satisfied).

Angular deformations (rotations)

θ_max ≤ θ_a; 0.037 ≤ 1° (satisfied).

Note. The allowable displacements were taken as v_a = 1.5 mm and θ_a = 1°.