Checking elements 5, 6 (upper column, body) with MDESIGN — example

1. Input data and results obtained

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

Meaning

Rigid Clamping – the fixed support; Position x – the position on the x axis; R_y – reaction force in the fixed support along the y axis; R_z – reaction force along the z axis; R – resultant reaction force in the transverse yz plane; R_x – reaction force along the x axis; Calculation results for point – the results calculated at point x; M_y – bending moment about the y axis; M_z – bending moment about the z axis; M_b – resultant maximum bending moment.

Bending moment diagrams

Note. The maximum bending moments are those at the support reaction (see above).

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).

2. Checks

Check under combined loading

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

σ_vmax ≤ σ_a,

or on safety factors,

c ≥ c_a,

where σ_vmax 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_vmax ≤ σ_a; 39.644 < 117.5 MPa (satisfied),

S_Fmin ≥ c_a; 5.928 > 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; 0.063 ≤ 1.5 mm (satisfied).

Angular deformations (rotations)

θ_max ≤ θ_a; 0.0423 ≤ 1° (satisfied).

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