Fundamental support and span moments

VOLUME A
The art of construction and detailing
Introduction
The structural frame
- Introduction
- Structural frame elements
  - Columns
  - Beams
  - Slabs
  - Staircases
  - Foundation
- STRUCTURAL FRAME LOADING
- BEHAVIOR OF THE STRUCTURAL FRAME
  - The behavior and reinforcement of a slab
  - Behavior and reinforcement of beams and columns
Construction methods
- The materials
- The moulds
- Thermal insulation of structural elements
- Concrete
- The steel
- Reinforcement specifications of antiseismic design
- Industrial stirrups - stirrup cages
- Standardized cross-sections of reinforced concrete elements
Reinforcement
- Columns
- Shear walls
- Composite elements
- Beams
- Slabs
- Staircases
- Foundation
Quantity surveying
- Estimation of the concrete’s quantity
- Estimation of the formworks’ quantity
- Estimation of the spacers’ quantity
- Estimation of the reinforcements' quantity
- Total estimation of the materials’ quantities
- Optimization of the reinforcement schedule
- Estimation of the structural frame’s cost
- Electronic exchange of designs - bids - orders
Detailing drawings
- General
- The drawings’ title block
- Carpenter’s drawings
- Steel fixer’s drawings
Tables
- Table 1
- Table 2
- Table 3
Drawings
- Carpenter’s drawings
- Steel fixer’s drawings
Model (exemplary) construction

« Shear forces and support reaction forces Bending moments in continuous two-way slabs »

Fundamental support and span moments - Deflection curves

Two calculation methods are: Marcus method and Czerny method.

MARCUS method

In this method, the slab is replaced by two intersecting strips along directions x and y respectively, which meet in its middle point m.

Depending on the support conditions, the strips are classified as simply supported, fixed at one end or fixed at both ends.

Each strip, subjected to its corresponding load, provides an elastic line and a bending moment diagram:

Simply supported strip

Strip fixed at one end

Figure 4.6.2.1: Strip fixed at both ends

Each strip bears load p_x ή p_y such as:

Loads p_x=k_x·p and p_y=k_y·p (k_x+k_y=1.00) are calculated according to equations (1) and (2), where:

[*]

Notef⁰_m=c_x·p_x·l⁴_x/(E·I)=12·c_x·k_x·l⁴_x/(E·b·h³)=a^o_m·l⁴_x/(E·1.0·h³), where a ^o_m=12·c_x·k_x

The bending moments on the spans and the supports are calculated by analyzing simply supported, fixed at one end or fixed at both ends strips, subjected to load p_x and p_y in both directions respectively.

The consideration of isolated intersecting strips ignores the effect of connecting adjacent strips through twisting moments. The favourable effect of the twisting moments on the span bending moments is provided by the twist coefficients ν_i (i=x, y), where:

The twisting moments affect only the span bending moments.

When the slab supports are not monolithic, such as when coincident free edges exist and no twist reinforcement is provided, then the resulting bending moments on the spans are greater, i.e.:

The equations for the six cases of two-way slabs are specialized according to the support types. Indicatively, coefficients for three aspect ratios are determined:

Type 1 (four simply supported edges)

For ε=2/3: k_x=0.165, k_y=0.835, ν_x=0.691, ν_y =0.691, 100a_m=1.78 (against 9.27/1.5⁴=1.83) [*]NoteResulting values in the example are compared with the corresponding of the differential equations of elasticity theory of slabs, for the same aspect ratio.

For ε=1.0: k_x=0.500, k_y=0.500, ν_x=0.583, ν_y=0.583, 100a_m=4.55 (against 4.87)

For ε=3/2: k_x=0.835, k_y=0.165, ν_x=0.691, ν_y=0.691, 100a_m=9.02 (against 9.27)

Type 2 (one fixed edge)

For ε=2/3: k_x=0.073, k_y=0.927, ν_x=0.863, ν_y=0.807, 100a_m=0.98 (against 4.98/1.5⁴=0.98)

For ε=1.0: k_x=0.286, k_y=0.714, ν_x=0.762,ν_y=0.665, 100a_m=3.41 (against 3.34)

For ε=3/2: k_x=0.669, k_y=0.331, ν_x=0.752, ν_y=0.651, 100a_m=7.86 (against 7.73)

Type 3 (two opposite fixed edges)

For ε=2/3: k_x=0.038, k_y=0.962, ν_x=0.929, ν_y=0.881, 100a_m=0.55 (against 2.97/1.5⁴=0.59)

For ε=1.0: k_x=0.167, k_y=0.833, ν_x=0.861, ν_y=0.769, 100a_m=2.25 (against 2.30)

For ε=3/2: k_x=0.503, k_y=0.497, ν_x=0.814, ν_y=0.690, 100a_m=6.40 (against 6.39)

Type 4 (two adjacent fixed edges)

For ε=2/3: k_x=0.165, k_y=0.835, ν_x=0.826, ν_y=0.826, 100a_m=0.85 (against 4.59/1.5⁴=0.91)

For ε=1.0: k_x=0.500, k_y=0.500, ν_x=0.765, ν_y=0.765, 100a_m=2.39 (against 2.52)

For ε=3/2: k_x=0.835, k_y=0.165, ν_x=0.826, ν_y=0.826, 100a_m=4.31 (against 4.59)

Type 5 (three fixed edges)

For ε=2/3: k_x=0.090, k_y=0.910, ν_x=0.905, ν_y=0.888, 100a_m=0.51 (against 2.9/1.5⁴=0.57)

For ε=1.0: k_x=0.333, k_y=0.667, ν_x=0.844, ν_y=0.815, 100a_m=1.76 (against 1.88)

For ε=3/2: k_x=0.717, k_y=0.283, ν_x=0.850, ν_y=0.823, 100a_m=3.81 (against 4.09)

Type 6 (four fixed edges)

For ε=2/3: k_x=0.165, k_y=0.835, ν_x=0.897 and ν_y=0.897, 100a_m=0.46 (against 2.64/1.5⁴=0.52)

For ε=1.0: k_x=0.500, k_y=0.500, ν_x=0.861 and ν_y=0.861, 100a_m=1.35 (against 1.52)

For ε=3/2: k_x=0.835, k_y=0.165, ν_x=0.897 and ν_y=0.897, 100a_m=2.34 (against 2.64)

Notes:

In case of ribbed (Zoellner) slab, moments and deflection curve are calculated approximately by using reduced factor ν_x, ν_y e.g. equal to ν'_x, ν'_y, or even with value 1.0.
To calculate the deflection the actual moments of inertia I of the ribbed slab is used, obtained by multiplying the corresponding table value with the reduction factor Ι/(1.0· h3/12)<1.0.

CZERNY's elasticity method

Figure 4.6.2.2

The coefficients used for the determination of spans and supports bending moments result from the tables of §4.6.1.2, depending on the aspect ratio ε = l_x / l_y (wherein l_x is the shortest dimension) and the type of support. Thus:

Notes:

Czerny method, the bending moments and the shear forces are expressed as a function of the shortest distance lx, whereas in Marcus method, the bending moment in each direction is a function of the respective dimension.
Slab tables usually referred to aspect ratios l_y/l_x≥1. When l_y/l_x<1, slabs are considered rotated by 90 ^ο.
Twisting moments M_xy are also provided by certain tables (when proper conditions met), as well as shear forces due to bending with or without twisting moments at the edges.

« Shear forces and support reaction forces Bending moments in continuous two-way slabs »