Civil Engineering

CE 334 Exam Resources Fall 2019 

DESIGN AIDS & CHARTS 

Reinforcing steel 

Bar # Diameter (in) Area (in2)

3 0.375 0.11

4 0.500 0.20

5 0.625 0.31

6 0.750 0.44

7 0.875 0.60

8 1.00 0.79

9 1.13 1.0

10 1.27 1.27

11 1.41 1.56

14 1.69 2.25

18 2.26 4.00

    Minimum Concrete Cover 

Concrete exposure Member Reinforcement Specified cover, in. Cast against and permanently in contact with ground All All 3

Exposed to weather or in contact with ground All

No. 6 through No. 18 bars 2

No. 5 bar, W31 or D31 wire, and smaller 1-1/2

Not exposed to weather or in contact with ground

Slabs, joists, and walls

No. 14 & No. 18 bars 1-1/2

No. 11 bar & smaller 3/4

Beams, columns, pedestals, and tension ties

Primary reinforcement, stirrups, ties, spirals, and hoops 1-1/2

 

ACI Moment and Shear Coefficients 

Moment Location Condition Mu

Positive End span

Discontinuous end integral with support wuℓn 2 /14

Discontinuous end unrestrained wuℓn 2 /11

Interior spans All wuℓn2/16

Negative [1]

Interior face of exterior support

Member built integrally with supporting spandrel beam wuℓn

2 /24

Member built integrally with supporting column wuℓn

2 /16

Exterior face of first interior support

Two spans wuℓn 2 /9

More than two spans wuℓn 2 /10

Face of other supports All wuℓn2/11

Face of all supports satisfying (a) or (b)

(a) slabs with spans not exceeding 10 ft (b) beams where ratio of sum of column stiffnesses to beam stiffness exceeds 8 at each end of span

wuℓn 2 /12

Shear Exterior face of first interior support All

1.15wuℓn/2

All other supports wuℓn/2  

 

 

T Beam Overhangs Effective Flange Width 

Flange location

Effective overhanging flange width, beyond face of web

Each side of web Least of:

8h

sw/2

ℓn/8

One side of web Least of:

6h

sw/2

ℓn/12

 

Minimum Thickness of BEAMS AND SLABS for no Deflection Calculations 

Support condition BEAM Minimum h[1]

SLAB Minimum h[1]

Simply supported ℓ/16 ℓ/20 One end continuous ℓ/18.5 ℓ/24 Both ends continuous ℓ/21 ℓ/28 Cantilever ℓ/8 ℓ/10  

 

 

 

 

 

Coefficient of Resistance Rn 

∅  

EQUATIONS 

Basic Load Combinations 

U = 1.4D  U = 1.2D+1.6L    U = 1.2D+1.6L+0.5(Lr or S or R)    U = 1.2D+1.0W+1.0L+0.5(Lr or S or R) 

Basic Design Equations 

φMn ≥ Mu  φVn ≥ Vu  φPn ≥ Pu  φTn ≥ Tu 

Strength Reduction Factors 

φ = 0.9 for tension controlled sections when εt ≥ 0.005 

φ = 0.65 for compression controlled sections when εt ≤ εy 

φ is linearly interpolated when εt is between εy and 0.005  φ = 0.65 + (εt – 0.002)(250/3) for fy = 60 ksi   

φ = 0.75 for shear and torsion 

φ = 0.65 for tied columns    φ = 0.75 for spiral columns 

Concrete Modulus of Elasticity 

Ec = 33(wc1.5)   = 57,000  for normal weight concrete 

Concrete Density 

Normal weight reinforced concrete = 150 lb/ft3    Normal weight unreinforced concrete = 144 lb/ft3 

Concrete Tensile Strength 

Modulus of rupture fr = 7.5λ  

Splitting (split cylinder) tensile strength fct = 6.7λ  

Lightweight Concrete Factor 

Normal weight λ = 1.0    sand lightweight λ = 0.85    all lightweight λ = 0.75 

Cracking Moment 

Mcr =   

Moment Strength 

T = Asfy     C = 0.85f’cAc    T = C to find depth of stress block a  Asfy = 0.85f’c ab if stress block is rectangular

Mn = T x jd = C x jd      jd = d – a/2 if stress block is rectangular 

Neutral axis location c = a/β1    εt from strain linearity  εt =  εcu  εcu = 0.003 

β1 = 0.85 for f’c ≤ 4000 psi  β1 = 0.65 for f’c ≥ 8000 psi  β1 = 0.85 – 0.05  between 4000 and 8000 psi 

Minimum Area if Steel in a Beam Section 

As, min  =  bwd   3  not less than 200 psi   

Statically determinate T beams with flange in tension use smaller of be or 2bw for bw in the As, min equation 

One Way Slabs – Design Based on a 12” Wide Section (b=12”) 

Spacing of reinforcement   =    or s = 12

Shrinkage and temperature reinforcement: required perpendicular to main steel   provided for moment strength 

As&t = 0.0018bh for Grade 60  As&t = 0.002bh for Grade 40  As&t = 0.0018bh ,  ≥ 0.0014 for higher strengths 

As, min for main steel = As&t  

Spacing limits are the smaller of 3h or 18” for main steel and the smaller of 5h or 18” for temperature & shrinkage steel 

Shear  Equations 

φVn = φVc + φVs   φ= 0.75 for shear 

Vc = 2λ  

Vs =  

Av min =  . but   0.75   ≥ 50 psi 

Vs max = 8   Maximum stirrup spacing = d/2 or 24” when Vs ≤ 4       d/4 or 12” when Vs ≥ 4       Stirrups required if Vu ≥ ½ φVc for most members 

Stirrups not required if Vu ≤ φVc for slabs and for beams with h ≤ 10” 

Maximum shear for design of beams is at distance = d from face of support if there is a compression reaction at support 

Shear from pattern loading at midspan of uniformly loaded beams Vu, midspan = 

Development of Reinforcement Equations 

     but not less than 12”  not greater than 2.5 

1.3 for bars with 12” of fresh concrete cast below them     = 1.5 for epoxy coated bars unless cover > 3d  and clear spacing > 6db when 1.2 can be used 

0.8 for No. 6 bars and smaller   but not less than 8db or 6” 

For standard hooks,   = 1.2 when rebar is epoxy-coated

 = 0.7 for No. 11 bars and smaller with side cover ≥ 2.5 

 = 0.8 for No. 11 bars and smaller with ties or stirrups placed along the hooked bar at spacing ≤3db 

Critical sections for development of bars is at points of maximum moment and at the cut off point for continuing bars. 

Actual cut off point must extend d or 12db beyond the theoretical cut off point 

For positive moment bars: 

At least 1/3As must extend at least 6” into a simple support and at least 1/4As must extend 6” into a continuous support 

At simple supports with a compression reaction,  1.3  

At inflection points,   = distance bars extend beyond support centerline at a simple support, or the larger of d, 12db at an inflection point For negative moment bars: 

At least 1/3As must extend the largest of d, 12db or   past the inflection point into the positive moment zone

Limits on Crack Widths 

Bar spacing s ≤  15 ( , 2.5  but not greater than 12 ( ,   where fs can be approximated as 2/3fy Deflection Calculations 

Ieff =  1 δLL = δDL+LL ‐ δDL 

δLT = δLL + λΔ δDL + λΔ δSL

λΔ=

Time-Dependent Factor for sustained loads Sustained load duration, months

Time-dependent factor ξ

3 1.0

6 1.2

12 1.4

60 or more 2.0

Maximum permissible calculated deflections

Member Condition Deflection to be considered Deflection limitation

Flat roofs Not supporting or attached to nonstructural elements likely to be damaged by large deflections

Immediate deflection due to maximum of Lr, S, and R ℓ/180

[1]

Floors Immediate deflection due to L ℓ/360

Roof or floors

Supporting or attached to nonstructural elements

Likely to be damaged by large deflections

That part of the total deflection occurring after attachment of nonstructural elements, which is the sum of the time-dependent deflection due to all sustained loads and the immediate deflection due to any additional live load[2]

ℓ/480[3]

Not likely to be damaged by large deflections

ℓ/240[4]

Ieff = Ig when Ma < Mcr

   

NOTATION 

a = depth of equivalent rectangular stress block, in. Ab = area of an individual bar or wire, in.2 Ac = area of concrete compression zone, in.2 Ag = gross area of concrete section, in.2 As = area of nonprestressed longitudinal tension reinforcement, in.2 As′ = area of compression reinforcement, in.2 As,min = minimum area of flexural reinforcement, in.2 Atr = total cross-sectional area of all transverse reinforcement within spacing s that crosses the potential plane of splitting through the reinforcement being developed, in.2 Av = area of shear reinforcement within spacing s, in.2 Av,min = minimum area of shear reinforcement within spacing s, in.2 b = width of compression face of member, in. bf = width of flange, in. be = effective flange width of T section, in. bw = web width, in. c = distance from extreme compression fiber to neutral axis, in. cb = lesser of: (a) the distance from center of a bar or wire to nearest concrete surface, and (b) one-half the center-to-center spacing of bars or wires being developed, in. cc = clear cover of reinforcement, in. d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement, in. d′ = distance from extreme compression fiber to centroid of longitudinal compression reinforcement, in. dagg = nominal maximum size of coarse aggregate, in. db = nominal diameter of bar, wire, or prestressing strand, in. D = effect of service dead load Ec = modulus of elasticity of concrete, psi EI = flexural stiffness of member, in.2-lb (EI)eff = effective flexural stiffness of member, in.2-lb Es = modulus of elasticity of reinforcement steel, psi fc′ = specified compressive strength of concrete, psi

cf  = square root of specified compressive strength of concrete, psi fr = modulus of rupture of concrete, psi fs = tensile stress in reinforcement at service loads, psi fy = specified yield strength for nonprestressed reinforcement, psi fyt = specified yield strength of transverse reinforcement, psi h = overall thickness, height, or depth of member, in. I = moment of inertia of section about centroidal axis, in.4 Icr = moment of inertia of cracked section transformed to concrete, in.4 Ie = effective moment of inertia for calculation of deflection, in.4 Ig = moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, in.4 jd = moment (lever) arm separating internal tension force T from internal compression force C, in. Ktr = transverse reinforcement index, in. ℓ = span length of beam or one-way slab; clear projection of cantilever, in. ℓa = additional embedment length beyond centerline of support or point of inflection, in. ℓd = development length in tension of deformed bar, deformed wire, plain and deformed welded wire reinforcement, or pretensioned strand, in. ℓdc = development length in compression of deformed bars and deformed wire, in. ℓdh = development length in tension of deformed bar or deformed wire with a standard hook, measured from outside end of hook, point of tangency, toward critical section, in. ℓn = length of clear span measured face-to-face of supports, in. L = effect of service live load Lr = effect of service roof live load Ma = maximum moment in member due to service loads at stage deflection is calculated, in.-lb

Mcr = cracking moment, in.-lb Mmax = maximum factored moment at section due to externally applied loads, in.-lb Mn = nominal flexural strength at section, in.-lb Mu = factored moment at section, in.-lb Nu = factored axial force normal to cross section occurring simultaneously with Vu or Tu; to be taken as positive for compression and negative for tension, lb n = number of bars or wires being developed along the plane of splitting n = modular ratio = Es/Ec Pc = critical buckling load, lb Pn = nominal axial compressive strength of member, lb Pn,max = maximum nominal axial compressive strength of a member, lb Pnt = nominal axial tensile strength of member, lb Pnt,max = maximum nominal axial tensile strength of member, lb Po = nominal axial strength at zero eccentricity, lb Pu = factored axial force; to be taken as positive for compression and negative for tension, lb PΔ = secondary moment due to lateral deflection, in.-lb s = center-to-center spacing of items, such as longitudinal reinforcement, transverse reinforcement, tendons, or anchors, in. sw = clear distance between adjacent webs, in. tf = thickness of flange, in. U = strength of a member or cross section required to resist factored loads or related internal moments and forces in such combinations as stipulated in this Code Vc = nominal shear strength provided by concrete, lb Vn = nominal shear strength, lb Vs = nominal shear strength provided by shear reinforcement, lb Vu = factored shear force at section, lb wc = density, unit weight, of normalweight concrete or equilibrium density of lightweight concrete, lb/ft3 wu  =  factored load per unit length of beam or one‐way slab, lb 

β1 = factor relating depth of equivalent rectangular compressive stress block to depth of neutral axis δDL = immediate deflection from service dead load δLL = immediate deflection from service live load δSL = immediate deflection from sustained live load δLT = long-term deflection εcu = concrete crushing strain εt = net tensile strain in extreme layer of longitudinal tension reinforcement at nominal strength, excluding strains due to effective prestress, creep, shrinkage, and temperature εty = value of net tensile strain in the extreme layer of longitudinal tension reinforcement used to define a compression-controlled section λ = modification factor to reflect the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength λΔ = multiplier used for additional deflection due to long-term effects ξ = time-dependent factor for sustained load ρ = ratio of As to bd ρ′ = ratio of As′ to bd ρw = ratio of As to bwd ϕ = strength reduction factor ψe = factor used to modify development length based on reinforcement coating ψr = factor used to modify development length based on confining reinforcement ψs = factor used to modify development length based on reinforcement size ψt = factor used to modify development length for casting location in tension