The Lugeon Test: Methodology, Interpretation, and Applications in Geotechnical Engineering

 Here’s a formula sheet for Design of Steel Structures in a clear table format, useful for quick revision before exams or viva. The formulas follow standard steel design concepts (IS 800:2007 / AISC basis):

Topic	Formula	Description / Notes
Axial Stress	$\sigma = \dfrac{P}{A}$	Axial stress in tension/compression, where $P$ = load, $A$ = area
Slenderness Ratio	$\lambda = \dfrac{L_{eff}}{r}$	$L_{eff}$ = effective length, $r = \sqrt{I/A}$ = radius of gyration
Euler’s Buckling Load	$P_{cr} = \dfrac{\pi^2 E I}{(L_{eff})^2}$	Critical load for long slender columns
Bending Stress (Flexural Formula)	$\sigma_b = \dfrac{M y}{I}$	$M$ = bending moment, $y$ = distance from N.A., $I$ = moment of inertia
Shear Stress	$\tau = \dfrac{VQ}{I b}$	$V$ = shear force, $Q$ = first moment of area, $b$ = width
Plastic Section Modulus	$Z_p = \sum A_i y_i$	Used in plastic design, $y_i$ = distance of centroid of area $A_i$ from plastic neutral axis
Elastic Section Modulus	$Z_e = \dfrac{I}{y_{max}}$	Ratio of moment of inertia to extreme fiber distance
Bending Strength	$M = f_b Z$	Design moment capacity, $f_b$ = permissible/design bending stress
Torsional Shear Stress	$\tau = \dfrac{T r}{J}$	$T$ = torque, $r$ = distance to outer fiber, $J$ = polar moment of inertia
Torsional Angle of Twist	$\theta = \dfrac{T L}{G J}$	Angle of twist for shaft/beam under torsion
Deflection of Simply Supported Beam (UDL)	$\Delta_{max} = \dfrac{5 w L^4}{384 E I}$	$w$ = load per unit length
Deflection of Simply Supported Beam (Point Load)	$\Delta_{max} = \dfrac{P L^3}{48 E I}$	Midspan deflection for central load
Shear Capacity of Bolt (Single Shear)	$V = \dfrac{\pi d^2}{4} \cdot \dfrac{f_{ub}}{\sqrt{3}\,\gamma_{mb}}$	$d$ = bolt dia., $f_{ub}$ = ultimate stress
Bearing Capacity of Bolt	$V_{bearing} = 2.5 k d t f_u / \gamma_{mb}$	$t$ = plate thickness, $k$ = hole factor
Weld Strength (Fillet)	$P = 0.7 t f_w L$	$t$ = throat thickness, $f_w$ = weld strength, $L$ = weld length
Effective Length of Column	$L_{eff} = K L$	$K$ = effective length factor (depends on end conditions)
Shear Strength of Web (No Stiffener)	$V_c = \dfrac{d t_w f_y}{\sqrt{3} \gamma_{m0}}$	$d$ = depth of web, $t_w$ = web thickness
Plate Girder Web Buckling Check	$\lambda_w = \dfrac{d}{t_w} \sqrt{\dfrac{f_y}{250}}$	Used to check need for stiffeners
Design Strength in Tension (Gross)	$T_{dg} = \dfrac{A_g f_y}{\gamma_{m0}}$	$A_g$ = gross area
Design Strength in Tension (Net)	$T_{dn} = \dfrac{A_n f_u}{\gamma_{m1}}$	$A_n$ = net area

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