Software Knowledge Base

Note ID: 000005 – Applications: R/C Building

Title: Deep Beam and Wall Design Module

Description: Detailed outline of the Deep Beam and Wall Design Module within R/C Building, including a step-by-step practical example

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Date Created: 01/11/2007

Date Modified: 01/11/2007

The Concept

Deep beams and shear walls are widely used structural elements and because they can support weight from several floors and they can span larger distances between the supports without any beams, they are an especially efficient type of construction. R/C BUILDING software offers a set of state-of-the-art analytical tools, based on the Finite Element approach that can be used in to analyse, design and detail deep beams and shear walls.

The main advantage of the Finite Element approach is that we can get a very good insight into the distribution of the internal forces, which will allow us to distribute the material in a manner that will reduce the cost and at the same time to significantly increase the overall safety.

The walls and deep beams interact with other structural elements on the same floor and with the elements on the upper and lower levels. Therefore a 3D structural model is the most appropriate approach to analyse deep beams. The design of deep beams is governed by the internal stress distribution, and it cannot be performed by the simplified beam theory based on uni-axial state of stresses. The most accurate approach for deep beam design is to evaluate the distribution of in-plane stresses, and to introduce steel bars where the concrete tension strength is exceeded. The stress distribution in a deep beam is a bi-axial (2D) state of stress, where all in-plane stresses are considered. In the R/C BUILDING software all componential stresses ( σ_x σ_y τ_xy ) and the principal stresses (σ₁ and σ₂ ) are obtained by the Finite Element Method (FEM). The stresses are evaluated in a number of detailed points in the wall (see Fig 1 below). The load bearing capacity of the wall is provided by limiting the compression stresses in the concrete and the tensile stresses in the reinforcing steel.

The design is based on two rules:

1. When the principal stress in tension σ₁ is greater than the tensile strength of the concrete, cracks will form and the tensile stress will be taken by the reinforcing bars in the horizontal and the vertical direction. If the stress in the bars is limited to a specific magnitude, then the crack width can be controlled, which in turn will ensure the strength and serviceability of the wall.

2. When the principal stress in compression σ₂ is greater than the compressive strength of concrete, it is assumed that the wall will fail. In this case the thickness of the wall or the concrete grade should be increased. In practical designs this should be a rare case.

Fig 1. - Rectangular grid of design cells xd/yd, and principal stresses

The wall is sub-divided into a number of rectangular design cells (xd/yd). The calculation of the required reinforcement is performed at the middle points of each design cell (xd/yd). The cell size, “xd” and “yd”, is between 0.2m and to 0.3m (see Fig. 1 above). The design procedure consists of calculating the vertical and the horizontal components of the total force acting at each cell, taking into account the thickness of the wall “t” and the orientation of the tensile principal stress, angle alpha “α”. Note that the steel design is based on the principle stresses ( σ₁ and σ₂ ), while the shear stress is zero. The angle alpha “α” is defined between the positive X axis and the direction of the principle stress σ₁ , anti-clockwise (see Fig. 1).

If a wall is subjected to out-of-plane actions, such as out-of-plane bending, then the stress-strain state becomes three-dimensional and additional design considerations are needed. The three-dimensional stress state is not considered in this method.

Theoretical Approach

The first step in the analysis is to evaluate the principal stress in tension σ₁ at the centre of each rectangular design cell (xd/yd). If the principal stress in tension is greater than the tensile strength of the concrete (as defined by the user), then this is an indication that in that cell some reinforcement is required. The amount of required reinforcement will be determined in such a manner to limit the tensile stress in the bars, and this in turn will limit the crack width. For example, we if we assume the tensile stress in 12mm (f_sy = 500MPa) reinforcing bars is limited to 125 MPa, and the crack width will be less than 0.3mm. The procedure for calculating the reinforcement is given below.

For each design cell (xd/yd) the total tensile horizontal force H (kN), and the total tension vertical force V (kN) are calculated as follows:

(1a)

(1b)

The total required cross section area for the horizontal and the vertical reinforcement in the design cell xd/yd is calculated as follows:

(2a)

(2b)

The user may adopt a basic reinforcement grid in the horizontal and vertical directions H_bg and V_bg. This basic steel grid will be taken into account in the field xd/yd with the expressions:

(3a)

(3b)

So that the additional reinforcement can be calculated based on the difference between the total required steel and the basic grid for the horizontal and vertical bars:

(4a)

(4b)

If A_{dif_h} is less than or equal to zero, it means that the horizontal basic grid is sufficient and that no additional horizontal reinforcement is required. Similarly, if A_{dif_v} is less than or equal to zero, it means that the vertical basic grid is sufficient and that no additional vertical reinforcement is needed. Note that the software then assumes that the basic grid (as defined by the user) is adopted throughout the entire area of the wall.

For the cases where A_{dif_h} or A_{dif_v} are greater than zero, some additional reinforcement in the horizontal and/or vertical direction is evaluated. The software will attempt to place a single additional bar between the bars of the basic grid, starting with a bar diameter of 12mm. If this 12mm bar is not sufficient, the software will increase the bar diameter to 16mm and finally to 20mm. If a 20mm bar is not sufficient, the software will indicate the number of bars that are required in-between each basic grid spacing, as shown in Fig 2.

Fig 2 – Basic Grid and Additional Bars

All additional bars are same size. The uses can always check the amount of additional steel area (mm²) required in-between the basic grid bars by moving the mouse over the wall.

If the stresses exceed the concrete tensile strength in one design cell, the required reinforcement can be spread over three cells, rather than one. This option is available under the “Smooth Extra Bars” switch.

The parameters definitions contained within this Chapter are listed as follows:

V_bg – Given vertical basic grid (in mm²/m')

H_bg – Given horizontal basic grid (in mm²/m')

S_max – Maximum allowed spacing in both the vertical and horizontal directions (in mm)

D_h – Selected horizontal bar diameter (in mm)

D_v – Selected vertical bar diameter (in mm)

n_L - Number of layers for distribution of the reinforcement (usually n_L=2)

σ_a – Allowable stress in the reinforcement (N/mm²)

σ₁ – Principal tensile stress in the concrete (kN/m²)

α – Angle of orientation of the principal tensile stress (degrees)

t – Membrane thickness of the element (m)

xd – Horizontal length of the considered rectangular field (m)

yd – Vertical length of the considered rectangular field (m)

Software Usage

The first step is to mesh and analyse an R/C BUILDNG model with the “Wall FE Advanced” option switched on. This setting is available under the “Model and Solver Settings” input panel. For each wall the “Wall / Deep Beam” window can be opened from the main menu, under [Solve]. Once the window is opened, entering a wall number will bring up a particular wall element and we can then examine all the results and design the wall.

Before performing the design, we need to enter some values for the concrete compressive strength, concrete tensile strength, as well as the anchoring length. R/C BUILDING software offers some default values, but the designer may change these values at any time. The design process can be performed for any Load Combination including the Load Envelope.

As the design of the reinforcement is based on the allowable stresses in bars, the user may vary the value of the allowable steel stress σ_a , in order to examine the effect of this parameter on the final design. If for example, σ_a is equal to the yield strength of the reinforcement σ_y , the ultimate stresses concept is applied. (Note that the load combination factors should be in compliance with the ultimate stress concept.)

The user can also check the state of the principal stresses in tension and in compression for each Load Case and Load Combination (including Envelope Load Combination) by pressing the buttons labelled “Tension” or “Compression”. For each set of results it is possible to view the values of the compressive stresses for each design cell (xd/yd), by moving the mouse over the wall.

The orientation of the principal stresses is presented graphically by short lines in the middle of each design cell. These lines will represent the stress “flow” in the wall. If the principle stresses are exceeded, the software will indicate the relevant zones by displaying circles in a different colour. If red circles are displayed when the compression stresses are shown, this is an indication that either the wall thickness and/or the concrete grade need to be increased.

When the tensile stresses are displayed and grey circles are shown, this is an indication of the zones where some reinforcement will be needed. Once the stresses are examined the user may proceed with evaluating the reinforcement requirements by pressing the "Design" button.

The user can also check the distribution of the vertical and horizontal stresses along the bottom and top edges of the wall/deep beam element (using the buttons "Horiz. stress" and "Vert. stress" respectively).

Practical Considerations

R/C BUILDING software does not check buckling of the walls. Therefore the designer must consider the slenderness of the wall and assess the potential of buckling. If a deep beam is constrained at the floor levels (which makes the physical height of the wall about 3.0m) and if the wall thickness greater than 150mm, buckling should not be problem. However, the slenderness must be check manually for each deep beam design.

If a deep beam is supporting several levels of masonry walls within the R/C BUILDING model, it is recommended to reduce the modulus of elasticity of the wall above the deep beam by 4 to 6 times. This will guarantee that the wall above the deep beam will not take any significant load. This can be emphasised by increasing the modulus of elasticity of the deep beam by 2 or 4 times. Modifying the modulus of elasticity this was will “force” the deep beam to “work” harder and to attract larger internal forces. Increasing the modulus of elasticity of the deep beam can be used only for strength design. When examining deflection results, the modules of elasticity of the deep beam must not be increased.

The design of deep beam is governed by two main parameters:

- Concrete tensile strength.

- Steel allowable tensile stress.

The design engineer has to select both parameters. Normally, the tensile concrete strength can be evaluated by the formula in AS 3600, clause 6.1.1.3. For concrete grade 32MPa, the tensile strength can be calculated as 2.26 MPa. It is known that concrete can with stand much larger tensile stresses, such as 4, 6, or even 8 MPa. However these values are not reliable and therefore it is recommended to use a conservative value of about 2.0 MPa for the concrete tensile strength.

The allowable stress in the steel can be limited to 25% of the steel grade. For a steel grade of 500 MPa, the allowable stress can be taken as 125 MPa. This is considered as very conservative, which will limit the width of any cracks to 0.3mm. It is known that the steel stress can go up to 50% or even 100% of the steel strength without compromising the load bearing capacity of the deep beam. This figure has to be considered in conjunction with the values of the load factors. If we use the ultimate load factors of (1.2G and 1.5Q), we may allow a limit of up to 80% or 100% of the allowable tensile stress (500 MPa). We need to also limit the steel stresses to 25% for the service load combination (1.0G + 0.4Q), which will limit the crack width at service load conditions. A very conservative approach will be to limit the concrete tensile stress to 2MPa for the service load combination (1.0G + 0.4Q), which will prevent and cracks from forming in the concrete.

General Notes

Deep beams are typically made of concrete and they can support the load from several floors above the deep beam. It is not uncommon for a concrete deep beam to support the load form 10 to 15 levels. Sometimes the deep beams are made of grouted blocks. This type of construction should not be used for spans greater than 10m, and should not support more than 2 to 3 levels. If grouted block are used, we recommend that a very low value for the tensile strength of the masonry is adopted, for example 0.2 to 0.5 MPa. Also, special care must be taken during construction to ensure that all the cavities within the entire wall are completely filled with grout, and it is strongly recommended that a minimum reinforcement rate of 12@200 is adopted in both the vertical and the horizontal directions.

A concrete deep beam can span over larger distances such as 10 to 14m, but it must be supported by walls and/or columns at a minimum of two points. If a deep beam is not supported there will be large stress concentrations in the slab at the wall ends, which can easily cause local failure of the slab.

Normally the compression stress in the deep beams is not a problem. However, if the compression is larger than 50% of the concrete strength some special measures might be introduced. This can occur in small zones above the column supports, and in these cases it is recommended to extend the starter bars from the column into the wall and to place 3 to 5 stirrups in the wall. This approach will provide confinement of the concrete when high compression stresses are applied.

In a deep beam design the bars in the vertical direction are of equal significance as in the horizontal direction. It is recommended to adopt the same bar size and spacing for the vertical and horizontal bars. When using beam analogy, the final results will yield horizontal bars only in the bottom section of the beam, and this approach cannot be applied to the design of deep beams.

Usually the deformations of a deep beam are very small. We can make a simplified evaluation of the long term deflections by multiplying the short term deformations by a factor of 2 or 3.

The software does not check the minimum steel requirements for concrete wall elements according to AS3600, and this has to be checked manually by the design engineer. These requirements may include selecting a basic reinforcing grid that complies with the requirements in AS3600.

When a deep beam is used to transfer the load from the upper levels and is supported by columns, its primary function is to transfer (distribute) the load from the upper levels onto the supporting elements below. However the deep beam is also “holding” the slab attached to the bottom edge, which is “hanging” from the deep beam. The connection between the deep beam (wall) and the slab at the bottom edge is in tension (especially at the mid-span), and a series of anchors must be used to take this force. It is recommended to ignore the contribution of the concrete when designing these types of joints and to rely solely on the steel reinforcing bars.

Numerical Example

Let us consider a small building with a concrete deep beam that spans 12m. (see figure below)

Fig. 3 - Deep Beam Design Example

Once the model has been analysed, we can open the “Wall / Deep Beam Design” window and display the deflection results. We can then observe the deformed shape of the wall as shown in Fig. 4 below. The displacement magnitudes are very small and normally this does not govern the design.

Fig 4 - Displacement Results

The next step is to display and examine the compression stresses. We can observe that the maximum stresses are about 11 MPa, which is 30% of the concrete grade. We can also observe the stress “flow” forming an arch, as expected for this configuration of supporting conditions. (see Fig. 5)

Fig. 5 - Compression Stresses

We can now display the tensile stresses and we can observe that the stresses exceed the concrete tensile strength near the supports, which was set to 1.0 MPa in this example. (see Fig. 5)

Fig. 6 - Tensile Stresses, f’_ct = 1.0 MPa

If we try to design the reinforcement, assuming that f’_ct = 1.0 MPa and steel allowable stress of 125 MPa, we will are presented with some extra bars at both supports as shown in Fig. 7 below.

Fig. 7- Extra Bars Design, f’_ct = 1 MPa, steel allowable stress = 125 MPa

We can vary the values of f’_ct from 1 MPa to 2 MPa, and the allowable steel stress from 125 MPa to 250 MPa in order to examine the effect on the size and the number of extra bars required. (see Fig. 8)

In the case when f’_ct = 2 MPa and the allowable steel stress is increased to 250 MPa, only two extra bars (12mm diameter) are required at each support. This is considered to be the most realistic design.

In the case when f’_ct = 1 MPa and the allowable steel stress is limited to 125 MPa, there are several extra bars (20mm diameter) required at each support. This is considered to be the most conservative design.

Fig.8 - Extra Bars Design for Various f’_ct and Steel Allowable Stress

If this was a real design we would have adopted the number of extra bars as provided by the case when f’_ct = 1 MPa and an allowable steel stress of 125 MPa (using a bar diameter of 12mm).

This configuration will introduce a few more extra bars where required, however the material cost does not greatly increase while the factor of safety is almost doubled.

The above reasoning illustrates the advantage of the Finite Element approach, where we get a very good insight into the distribution of the internal forces, which will allow us to introduce a minimum amount of extra material exactly where it will make the largest contribution to the overall safety.

The next step is to consider the stresses along the top and bottom edge. We have already mentioned that apart from viewing the stresses inside the wall elements, it is necessary to check the vertical (normal) and horizontal (shear) stresses along the bottom and top edges of the deep beam. These stresses are used to calculate the required number of vertical anchors along the joint between the wall and the slab. The distribution of vertical stresses along the edges of this example is shown in Fig. 9 and the distribution of horizontal stresses (shear) is presented in Fig. 10.

Fig. 9 - Distribution of the vertical stresses along bottom and top edges

In Fig. 9 we can observe that the maximum vertical compression is at the supports (12 MPa) which can easily be taken but the concrete. The tensile zone on the bottom edge (not shown due to the scale) is concentrated in the middle of the wall (1.160 MPa). As this stress is less than 2 MPa, we may assume that theoretically the slab and wall will not separate along the bottom edge, however in practical design we always assume that this will be the case.

The horizontal edge stresses (shear) must also be considered as shown in Fig. 10. The shear stress may cause slippage of the wall relative to the slab along the already formed horizontal cracks that are caused by the tensile stresses (see Fig. 11).

To prevent these two failure modes we need to design vertical anchor bars along the wall edges with appropriate cross-section areas to resist pullout and slip actions. For these bars we should also provide appropriate an anchorage length.

Fig. 10 - Distribution of the horizontal stresses along bottom and top edges

Fig. 11 – Pullout and shear actions on the boundaries between the wall and slab

In order to design the anchors we should take into account the integral "I" section formed by the wall and the bottom and the top slabs. The anchors can be designed assuming that the steel cross-sectional area resists the total pullout force due to tensile vertical stresses, as well as the main part of the shear forces acting along the tensioned boundary zone.

For the wall in our example we assume that a crack between the wall and the slabs has formed, and that the entire stress will be taken by the anchors. We can design the necessary anchor bars on the bottom boundary between the wall and slab, as follows:

According with the vertical stress diagram (Fig. 9), the maximum vertical tensile stress in the bottom edge σ_{y_max} = 1.160 MPa. Therefore the total pullout force, N, to be resisted by the anchors will be:

N = 1,160 x 0.2 (stress multiplied by the wall thickness)

N = 232 kN/m

If we assume that we will anchor each vertical bar into the slab below using 2 layers of reinforcement, each with a rate of 12mm bars at a spacing of 300mm, we can evaluate the stress in the anchors as:

Steel area = 0.0007539 m² / m ( 753.9 mm² / m )

Stress in the bars = 232 / 0.0007539

Stress in the bars = 308 MPa < 500 MPa : OK for ultimate load

Note that the above calculations are applicable at the point of the maximum stress. We may average the stress by a distribution factor of 0.3 to 0.4, which will reduce the stress by 60% to 70%. We can confirm this by examining the principle stresses in tension along the bottom edge. The average bar stresses will be between 92 MPa to 123 MPa, which is less than 500 MPa.

The next step is to check the shear slip. The maximum shear stress of 4.454 MPa is at the wall ends near the supports (Fig. 10). Since this stress is concentrated over a very small zone we can use a distribution factor of 0.1 or 0.2. Therefore the average shear stress is:

0.2 x 4.454 MPa = 0.89 MPa < 2.0 MPa.

Usually, the shear stress will be resisted by the concrete along the wall edges. But if a crack between the wall and the slab develops, the shear will be taken by the anchors. Normally the shear stress is not of any significance, although it has to be checked in each design.

As a general rule we recommend to anchor each vertical bar from the wall into the slab. Then we have to check that the shear can be taken by the concrete. It is a very rare case when additional steel is needed to cover the shear between the wall and the slab.

The last step of this procedure is the calculation of the necessary anchorage length, which must be performed in compliance with the current design code AS 3600.