CIVIL ENGINEERING FORMULAS

42 CHAPTER TWO 0 0.05 0.10 0.15 m = MF WL W aL 0.20 0 1.0 0.1 0.9 0.2 0.8 G 2 = 0.05 G 2 = 0 G 2 = 0 G 2 = 0.15 G 2 G 2 = 0.20 = ...

BEAM FORMULAS 43 W = P W = wyL W = W = (n + 1)P nP yL xL xL yL w yL xL w X = 0 x = n y 1 + n G^2 = 0 S^3 = 0 G^2 = S^3 = 0 Case ...

44 CHAPTER TWO W = wyL 2 yL xL yL yL yL yL yL yL aL P PPPPPP yL xL xL Ldx xL yL y 0 w Case 6 G^2 =n y^2 = (^2) – 1 12 n+ 1 n– ...

BEAM FORMULAS 45 S^3  Pn/W. These values are given in Fig. 2.8 for some common types of loading. Formulas for moments due to de ...

46 CHAPTER TWO reciprocal deflections, we obtain the end moments of the deflected beam in Fig. 2.9 as (2.1) (2.2) In a similar m ...

BEAM FORMULAS 47 The mechanism method can be used to analyze rigid frames of constant sec- tion with fixed bases, as in Fig. 2.1 ...

L 2 L B B A E E L DA D CCP B C P (a) (b) Beam mechanism (c) Frame mechanism (d) Combination mechanism (e) (f) (g) AADDAD B P E B ...

BEAM FORMULAS 49 or if (2.6) A plastic hinge forms at this point when the moment equals kMP. For equilibrium, leading to (2.7) W ...

TABLE 2.1 Uniformly Loaded Continuous Beams over Equal Spans (Uniform load per unit lengthw;length of each spanl) Shear on eac ...

BEAM FORMULAS 51 on each side of the supports. Note that the shear is of the opposite sign on either side of the supports and th ...

52 CHAPTER TWO Maxwell’s theorem states that if unit loads rest upon a beam at two points, A andB, the deflection at Adue to the ...

BEAM FORMULAS 53 Shear at the end of a beam necessitates modification of the forms deter- mined earlier. The area required to re ...

Beam Fixed at One End, Load P Concentrated at Other End 1 2 Cross section B A B P P A l l b b x x y y^2 = x y y y h h 6 P bSs ...

B B z P A A P l l x b b z x b y y= x b y y y h h h h 6 P h^2 Ss 2 3 =k (const) h Rectangle: width (y) var -iable, depth (h) cons ...

B P B A x A l l x d y y^3 = x b y d y y h h 32 P Circle: πSs diam (y) variable Rectangle: width (b) con -stant, depth (y) variab ...

FIGURE 2.17 (Continued) B B A A x x l l y y y z h b b y= b b z y h h 3 Px^2 lSsh^2 Rectangle: width (z) var- iable, depth (h) co ...

B A P C x x l l y y d y^3 = x^2 x d b y y h h 16 P πlSs Rectangle: width (b) con- stant, depth (y) variable Elevation: two parab ...

FIGURE 2.17 (Continued) P C C AB b P x x P b y= x x b y x y 3 P Ssh^2 Rectangle: width (y) vari- able, depth (h) constant Rectan ...

OC P x AB b Rectangle: width (b) con- stant, depth (y) variable Plan: rectangle 3 Pl 4 bSs h= f= h y y h l 2 Elevation: ellipse ...

FIGURE 2.17 (Continued) x Rectangle: width (y) variable, depth (h) constant Plan: two parabolas with vertices at center of span ...