CIVIL ENGINEERING FORMULAS

CASE 3. Beam Supported Both Ends—Two Unequal Concentrated Loads, Unequally Distributed R = 1 L W (L – a) + W 1 b M =a L W (L – a ...

23 FIGURE 2.2 (Continued) CASE 4. Beam Supported Both Ends—Three Unequal Concentrated Loads, Unequally Distributed R = Wb + W 1 ...

D R 1 At center: At supports: Atx: Atx: R V V W x L M M 1 M (max) = (L^2 – 2Lx + x^2 ) WL 24 M 1 (max) = WL 12 L^2 6 M=–+Lx – x^ ...

FIGURE 2.2 (Continued) 25 CASE 6. Beam Fixed Both Ends—Concentrated Load at Any Point b^2 (3a + b) L^3 V (max) = Rwhena < b = ...

R 1 Loading At fixed end: At free end: Atx: Atx: Bending Shear D V W x L M M (max) = (x^4 – 4L^3 x + 3L^4 ) WL 2 M = Wx^2 2 L At ...

FIGURE 2.2 (Continued) 27 D R 1 V W x ab L M CASE 8. Beam Fixed at One End (Cantilever)—Concentrated Load at Any Point R 1 = V ( ...

Atx: when x < a Atx: when x > a Atx: when x > a Atx: when x = a = 0.414L Atx: when x < a Atx: when x > a Atx: whe ...

FIGURE 2.2 (Continued) 29 D At fixed end: Atx: Atx: when x = Atx: when x = 0.4215L Atx: R R 1 V W x L M M 1 M (max) = WL 9 128 M ...

Atx: when x < a Atx 1 : when x 1 < L V = R – w(a + x 1 ) Atx 1 : when x 1 < L Atx 2 : when x 2 < b V = w(b – x 2 ) A ...

FIGURE 2.2 (Continued) 31 D D VV R 1 Atx 1 : when x 1 < L Atx: when x < a Atx: when x < a R W 2 x aaL x 1 M (max) = (a ...

L Load x(l – x)wL^2 x(l – 2x^2 + x^3 ) xwL Shear Moment Elastic curve (a) w xL R= R R wL 2 R= wL 2 wL^2 8 wL^3 24 EI 5 wL^4 38 ...

33 P P P P L Load (1 – 2k)L PxL x(3k – 3k^2 – x^2 ) k(3x′ – 3x′^2 – k^2 ) k(3 – 4k^2 ) Shear Moment Elastic curve (e) xL (x < ...

P P Load Shear Moment Elastic curve (g) R 1 =L′P L L′ xPL′ xL x′L′ Px′L′ PL′ L R 2 = P R 2 R 1 L + L′ L 2(L + L′) – L′x′(1 + x′) ...

35 Load Shear Moment Elastic curve (j) R = wL wLx wL xL L w wL^2 2 wL^2 2 (3 – 4x + x^4 ) x^2 wL^4 24 EI wL^4 8 EI (^12) wL 2 Lo ...

Shear Moment (m) 0.5774l R 1 R 1 =V 1 R 2 =V2max Mmax ∆max ∆x atx = = 0.5774l Vx Mx R 2 V 1 l xW Mmax V 2 = W 3 W 3 =^2 W 3 = = ...

BEAM FORMULAS 37 Shear Moment (n) R=V Mmax(at center) ∆max(at center) ∆x R V R x W l V Mmax =(W l^2 – 4x^2 ) 2 l^2 =W 2 = Wl 6 W ...

38 CHAPTER TWO Shear Moment (p) R=V =P =Pl =Px = Mmax(at fixed end) ∆max(at free end) ∆x Mx R P V x l Mmax Pl^3 3 EI = P 6 EI (2 ...

P P R L LR AL LR LRLR R (a) (b) (c) ML ML M ML MR R MR kL VL VR kL kL k(l – k)PL = ++ k(l – k)PL BC FIGURE 2.4 Any span of a con ...

40 CHAPTER TWO R 1 d 1 y 11 d 1 =y 11 R 1 +y 12 R 2 +y 13 R 3 d 2 =y 21 R 1 +y 22 R 2 +y 23 R 3 d 3 =y 31 R 1 +y 32 R 2 +y 33 R ...

BEAM FORMULAS 41 kL kL kL kL kL k(l–k)^2 WL k(l–k)WL k(l–k)WL k^2 (l–k)WL W L L L (a) (b) (c) (d) w L WL 12 WL 12 L 2 WL W = w ...