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CIE525: Assignment 5
Strut and Tie Models
Manish Kumar
1.1
Problem Statement
A 26-inch deep precast concrete T-beam has a 12 in. thick web and carries factored end reactions
of 75 kips and 15 kips in the vertical direction and horizontal directions as shown in the figure
below. Load is transferred to the supporting corbel via a bearing plate that is 412 inches in
plan. Assume that the concrete compressive strength is 6000 psi and that ASTM Grade A615
Grade 60 rebar is used. Design and detail the reinforcement at the end of the beam using strut-
and-tie procedures. Draw elevations and sections that show all reinforcement and the bearing
plate. Your solution must include a drawing of the strut-and-tie model.
1.2 Solution
The solution process for analyzing concrete members, as outlined in Wight and MacGregor
(2009)and class notes (Whittaker, 2012), employed here is:
1. Identify and isolate the D (discontinuous) regions in the given member/structure
2. Draw an equivalent strut-and-tie model for the D-regions to transmit the forces
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Figure 2: Strut-and-tie model for the D-region
In the figure above, dashed line represents struts and solid line represents the ties.
1.2.3 Step 3: Compressive Strength of Nodal Zones and Struts
The compressive strength of the nodal zones is given by:
'0.85ce n cf f
Nodal zone A, E: Node A is a CCT node for which 0.8n . Hence effective strength of
concrete in the nodal zone A is given as '
0.85 0.85 0.8 6 4.08ce n cf f ksi .
Nodal zone B, F: This node is surrounded by three compressive struts and can be treated as
CCC, for which 1n .'
0.85 0.85 1 6 5.1ce n c
f f ksi
Nodal zone C, D: These nodes includes more than one tie, hence 0.6n ,
'0.85 0.85 0.6 6 3.06ce n cf f ksi
Struts AF, BF, BC, BE: These struts have room to expand and are assumed as bottle shaped
struts, which gives 0.75s ,'
0.85 0.85 0.75 6 3.8ce s cf f ksi
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1.2.6 Step 6: Steel Required in the Ties
For the tie AB:
90AB
T k
2902
0.75 60
ust
y
FA in
f
2 #9 bars can be provided here. However, the development length for #9 bars is not available
here so an angle can be provided at the support and bars can be welded to the angle. The bearing
area of the angle would be based on the bearing strength of concrete of the concrete which
should not be less than nodal zone strength of A. The required bearing area is:
27522.6
0.65 0.85 6in
Required bearing area calculated through nodal zone strength is:
27524.5
0.75 4.08in
Hence area of2
24.5 in is required for the angle. A 3 3 1 / 2 angle is provided at the corner
along the 12"width of the beam.
For the tie CF:
100.4CF
T k
2100.42.23
0.75 60
ust
y
FA in
f
9 #5 bars can be used in three layers with 3 bars in each layer, as shown in Figure 4.Layers
should be equally spaced at 3" . Use #5 bars inside the top corners of the closed stirrups.
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Figure 4: Shear reinforcement provided for tie CF
For the tie CD:
As the force in CD is same to CF, it will have the same area requirement of2
2.23 in . 4 #8 bars
can be provided in the bottom which provides an equivalent force of 142.2 ( 100.4 )k k .
However, these need to be developed as well to acquire the full strength of the bars. The
development length for #8 bars is 54.8 inch, however available development length is only about
10 inches. So, either a mechanical anchor or U bars need to be provided.
The area of required U bars is:
2142.23.16
0.75 60sA in
Use 2 #8 U bars which bend adjacent to the end of the beam.
The actual effective width of tie is now calculated as bar diameter plus twice the cover, which is
equal to 4"tw
1.2.7 Adequacy of Strut-widths
It is necessary to find out widths of each strut and show that it will fit the given beam. Let us
assume a prismatic strut with thickness equal to the beams thickness of 12". In order to find out
the width, minimum of nodal zone compressive strength and corresponding strut compressive
strength will govern. Only strut AF and FE are checked, as they are expected to be critical.
Strut AF: Compressive strength of strut AF will govern 3.8cef ksi . Required width is:
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1063.1"
0.75 3.8 12w
Provided width is calculated as cos sin 4 cos 45 4 sin 45 5.6"o os t bw w l
Hence provided width is Ok!
Strut FE: Nodal zone strength of E will govern 4.08cef ksi , required width is:
1062.9"
0.75 4.08 12w
Beam can accommodate this width at top.
1.2.8
Crack Control
Longitudinal cracking can occur along the strut if the applied load is greater than twice the
cracking force 2n
C . Hence for strut AF:
'2 2 0.55 2 0.55 5.4 12 6 427n s cC w b f k
As factored load (106 k) is less than longitudinal cracking force in the strut, design is ok.
Also we have used 0.75s
assuming bottle shaped struts which require that adequate shear
reinforcement conforming to ACI 318 (ACI, 2011)must be provided, which requires:
sin 0.003si ii
A
bs
Let us provide two horizontal #4 U-shaped bars enclosing the strut AF, which gives:
4 0.2sin 45 0.0033 0.003
12 14
Hence provided design is OK!
1.3
Final Design
The final design of the dapped end of the beam is shown inFigure 5
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Figure 5: Design of the dapped end of the beam
References
ACI (2011). "Building code requirements for structural concrete and commentary." Report ACI
318-11, American Concrete Institute, USA.
Whittaker, A. S. (2012). "CIE 525: Concrete Design Class Notes."University at Buffalo, NY.
Wight, J. K., and MacGregor, J. G. (2009). "Reinforced concrete: Mechanics and design (5th
edition)." Pearson Prentice Hall, Upper Saddle River, NJ.