Structural Systems - Equations

Axial Stress

(maximum axial stress occurs along entire cross section) fa = P/A Axial Tension or Compression Stress (f-sub-a) = Axial Tension (P) / Area (A)

Bending Stress

(maximum bending stress occurs at the extreme fibers) fb = M/S Bending Stress (f-sub-b) = Moment (M) / Section Modulus (S) fb = Mc/I Bending Stress (f-sub-b) = Moment (M) x constant (c) / Moment of Inertia (I) (the greater the constant, the greater the bending stress)

Shear Stress

(maximum shear occurs at the neutral axis and is the same at both the vertical and horizontal axis) fv = 1.5V/A Shear Stress (f-sub-v) = 1.5 x Shear Force (V) / Area (A) fv = VQ/Ib Shear Strss (f-sub-v) = Shear Force (V) x Neutral Axis of area above plane (Q) / Moment of Intertia (I) x width of beam (b)

Seismic Response Coefficient

Cs = Sds/I/R Seismic Response Coefficient (C-sub-s) = Design Spectral Response at the short period of 0.2 sec (Sds) divided by the Response modification factor (R) based on the type of seismic force resisting system, divided by the importance factor (I) Response modification factor is laid out in a table, decided by building system/structural system

Modulus of Elasticity

E = f/ε Modulus of Elasticity (E) = Stress (f) / Strain (ε)

Force

F = Ma Force (F) = Mass (M) x Acceleration (a)

Foundation Pressure

F = P/A Foundation pressure (F) = load of the foundation (P) / Area (A)

Horizontal Force on a Retaining Wall

F = w²h/2 Force (F) = soil pressure (w) x height of wall (h)² / 2

Moment of a Point Load

M = PL/4 Moment (M) = Point Load (P) x length (L) / 4

Moment Equation

M = Pd Moment = Force (P) x distance (d)

Moment of a Uniform Load

M = wL²/8 Moment (M) = uniform load (w) x length (L)² / 8

Moment of combined Point Load and Uniform Loads

M = wL²/8 + PL/4

Moment of Inertia (three equations)

Moment of Inertia (occurs about the centroidal axis) I = bd³/12 Moment of Inertia (I) = base (b) x depth (d)³ / 12 Rectangle I = bd³/3 Moment of Inertia (I) = base (b) x depth (d)³ / 3 I-base = I + Ay² Moment of Inertia at base (I-base) = Moment of Inertia (I) + Area (A) x distance from centroid to base (y)₂

Neutral axis of area above Plane

Q = Ad Neutral axis of area above Plane (Q) = section area (A) x distance from centroid of rectangle to centroid of section above neutral axis (d)

Shear Diagram Shear Force

R = V = wL/2 Shear Resisting Force (R) = (V) = uniform load per foot (w) x distance (L) / 2

Section Modulus (three equations)

S = bd²/6 Section Modulus (S) = Base (b) x depth (d)²/6 S = M/Fb Section Modulus (S) = Moment (M) / Bending Stress (F-sub-b) S = I/c Section Modulus (S) = Moment of Inertia (I) / given constant

Yield Strength

The maximum stress that a structure or material can withstand without being permanently deformed. Fy (F-sub-y)

Lateral Earthquake Force

V = CsW Lateral force, or Base Shear = Seismic Response Coefficient (Cs) x total dead load (W)

Deflection (shortening of column or elongation of horizontal member)

e = PL/AE Deflection (e) = Force (P) x Length (L) / Area of cross section (A) x Modulus of Elasticity (E)

Stress

f = P/A Stress (f) = Total Force (P) / Area (A)

Thermal Equation (thermal stress in a restrained member)

ft = Ee∆t Thermal Stress (ft) = Modulus of Elasticity (E) x coefficient of thermal linear expansion (e) x temperature change (∆t)

Radius of Gyration

r = √I/A Radius of Gyration (r) = square root of the Moment of Inertia (I) / Area (A)

Slenderness Ratio

steel column: SR = KL/r Slenderness ratio (SR) = end condition (K) x unbraced length in inches (L) / radius of gyration (r) wood column: SR = KL/b Slenderness ratio (SR) = end condition (K) x unbraced length in inches (L) / cross section width of rectangle (b)

Strain

ε = e/L Strain (ε) = Deflection (e) / Original Length (L)

Deflection (of beam depth)

∆ = 5wL⁴/384EI Deflection (∆) = 5 x weight in lbs (w) x length (in feet x 12") (L)⁴ / 384 x Moment of Elasticity (E) x Moment of Inertia (I)

Thermal Equation (shortening or elongation due to temperature change)

∆ = eL∆t Thermal Change (∆) = coefficient of thermal linear expansion (e) x original length (L) x temperature change (∆t)