01i. Attachment I - Slab Capacity Check Report-AIMT Engineering 28June23.pdf

Slab Capacity Check of Building 20202 Kirtland Air Force Base

Dr. Mahmoud R. Taha, PE

June 28th, 2023

AIMT Engineering Service

632 Cedar St. NE, Albuquerque, NM 87106 www.aimteng.com

Contents

1. Introduction

2. Codes and design guidelines

3. Design Loads

4. Existing structure:

5. Material properties

6. Field inspection

7. Building model

7.1. Model of the as-built structure

7.2. Creep and shrinkage analysis

7.3. Cracked section check

8. Non-destructive testing of the building slab

8.1. Non-destructive testing results

8.2. Updated model and calculations based on the results of field testing

9. Deflection prediction

10. Allowable live load capacity

11. Lateral diaphragm capacity

11.1. Lateral transfer capacity of the original slab diaphragm per UBC 61-64

11.2. Lateral transfer capacity of the as-built slab diaphragm per IBC 2018

12. Repair/Strengthening Strategy

12.1. External Post-tensioning Design

12.2. Evaluation of selected repair strategy

12.3. Results of deflection analysis using post-tensioning

12.4. Ultimate capacity of the post-tensioned slab

13. Conclusions and Recommendations

14. Appendix

1. Introduction

Building 20202 at Kirtland Air Force Base (KAFB) is a concrete structure with perimeter and interior beams supporting a slab that spans continuously over the supports in the short direction of the building. This multistory building was constructed in the 1960s consisting of three isolated wings/segments with a basement located below the center segment of this facility. The structural elevated floor system is primarily a 4-1/2 inches thick slab and in isolated areas a 5-inch-thick slab. The concrete slab has been constructed with a 2,500-psi concrete mix. The flexural reinforcing is #4 alternate bent rebar at eight inches on center. Temperature and shrinkage reinforcing that runs perpendicular to the flexural reinforcement are #3 rebar at eighteen inches on center.

The elevated floors are that are constructed with a thickness of 4-1/2 inches and are experiencing service deflections that exceed the allowable deflections set forth in the IBC. The slab deflection, in each bay, is approximately 1” to 1-1/2 inches over a span of 12 feet eight inches as shown in Figure 1. This approximate deflection was measured with no live load in the bay. Work under this project is comprised of an investigative study required to determine the as-built floor slab capacity for the floors that currently do not meet the IBC standard for floor service deflections.

Verify the floor slab capacity utilizing the finite element model analysis (FEM) software. Two FEM are required; the first model is to determine the design capacity and service deflections that were anticipated with the original design. The second FEM model reflects the current as-built condition including but not limited to the floor deflection and the reduction in stiffness due to the primary restraint cracks along the perimeter and interior beams. The majority of elevated slabs of the building are all exhibiting similar conditions as the abated bay; therefore, the restraint cracks present should be expected throughout the building. This building will be utilized for office space with primarily fixed partition walls along column lines and lightweight prefabricated modular walls located within the slab bays.

Figure 1: Floor plan of Building 20202 at KAFB. The area marked is the main problem area where excessive deflection is observed.

2. Codes and design guidelines

The material properties such as elastic modulus, modulus of rupture, a moment of continuous slab at mid-section, cracking strain, etc. were estimated using ACI 318-14.

The lateral transfer capacity of the as-built slab diaphragm was evaluated per IBC2018 which references ASCE7-16.

The lateral transfer capacity of the original diaphragm was checked per UBC 1964 code.

The time-dependent effects of shrinkage and creep were calculated based on CEB-FIP model code 90 now adopted by most design codes worldwide.

The design compressive strength of in-place concrete based on core measurements was estimated from core strength results using ACI214.4R-03.

3. Design Loads

Table 1: Design loads and values used in the study.

Load case Formula/Value Dead load Geometry at 150 pcf density Live load (general floor space) 40 psf Live load (corridor space) 60 psf Superimposed dead load (pre-1980 renovations) 16 psf Superimposed dead load (post-1980 renovations):

These loads include.

1/2” gypsum board Suspended steel channel system ½” Asbestos slab

8.2 psf

2.2 psf

2.0 psf

4.0 psf

4. Existing structure:

Figure 2 shows the geometry of building 20202. Figure 3 and Figure 4 show the structural framing system including the beams and columns on the second floor respectively.

Figure 2: Rendering of a 3D computer model of building 20202 at KAFB

Figure 3: Structural framing system of the second floor in building 20202

Figure 4: Second-floor beam and slab framing in building 20202

5. Material properties

Table 2: Material properties of cast-in-place concrete

Material property Formula/Value 28-day compressive strength 𝑓 2500 psi Tensile strength 𝑓 7.5 𝑓 375 p𝑠𝑖 28-day modulus of elasticity 𝐸 57,000 𝑓 2,850 ksi

Table 3: Material properties for reinforcing steel

Material property Value Yield strength 𝑓 40 ksi Tensile strength 𝑓 60 ksi Expected yield strength 𝑓 44 ksi Expected Tensile strength 𝑓 66 ksi

6. Field inspection

A field inspection of the existing slab was performed measuring local deflections at mid-span with reference to the bottom of the existing columns at the floor elevation at multiple locations as shown in Error! Reference source not found..

Figure 5: Point locations of the deflection measured in the field. The area shown is at the second-floor center location (shaded area in Figure 1).

Deflection measurements at the designated location shown in Figure 5 are presented in Table 4.

It is evident that excessive deflection takes place at the floors and deflections ranging from 1.5 in to 3.0 in were measured.

Table 4: Slab deflections measured in the field.

Point A B C D E F G Inspection (in) 2.0 2.5 2.0 3.0 1.5 2.0 1.8

7. Building model

7.1. Model of the as-built structure

A finite element model using ETABS software was developed using the as-built drawings of Building 20202 KAFB analyzing the effects of self-weight alongside design loads consisting of 16 psf for superimposed dead load and live loads of 40 psf and 60 psf in the corridor. The resulting deflections were captured as shown in Figure with a maximum deflection of 0.27 in near the expansion joint between the center and secondary structures. Analysis of the effective stress and strains of the as-built slab in the center structure show a maximum stress value of 214 psi and a maximum strain of 0.00006 which satisfy design requirements as shown in Figure and Figure , respectively.

Figure 6: Deflections as measured in the as-built structure subject to service loads.

Figure 7: Elastic stress based on self-weight with maximum stress of 214 psi.

Figure 8: Elastic strain based on only self-weight with maximum strain of 0.00006.

7.2.Creep and shrinkage analysis

Both creep and shrinkage long-term deformations and strains were predicted for the concrete slab and supporting beams per structural drawings and according to CEB-FIP Model Code 90.

The following assumptions were made:

Age of structure = 59 years (Built in 1964) Relative humidity = 50% Age of concrete at beginning of shrinkage = 7 days

Age of concrete when the load was first applied (removal of formwork) = 7 days Type of cement: normal hardening cement Type of aggregate: normal weight aggregate

Calculations of additional strains (under self-weight) are provided in the appendix for the 4.5 in slab. The additional strain exerted from creep and shrinkage was calculated separately under self-weight and using an effective elastic modulus that accounts for the long-term time-dependent strain. Based on the 59 years structural age and given the average low humidity in Albuquerque, the maximum creep coefficient of 5.0 was calculated as shown in Figure . The shrinkage strain was also predicted assuming similar conditions assumed in creep prediction. Shrinkage prediction as shown in Figure . The results shown in the appendix indicate a creep stain value of

0.0004 and shrinkage strain of 0.0007. The effective elastic modulus accounting for the total strains including time-dependent strains was also calculated.

Figure 9: Evolution of the creep coefficient with time-based on FIB CEB model code 90.

Figure 10: Evolution of shrinkage strain with time according to FIB CEB model code 90.

Total Strain Elastic strain creep strain shrinkage strain 𝜀 𝜀 𝜀 𝜀 0.00116

𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒 𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑚𝑜𝑑𝑢𝑙𝑢𝑠,𝐸 𝑆𝑡𝑟𝑒𝑠𝑠 𝑓𝑟𝑜𝑚 𝑠𝑒𝑙𝑓 𝑤𝑒𝑖𝑔ℎ𝑡,𝜎

𝑇𝑜𝑡𝑎𝑙 𝑠𝑡𝑟𝑎𝑖𝑛, 𝜀

𝐸 168,000 psi

7.3.Cracked section check.

Based on ACI 318-14, the modulus of rupture (or cracking strength) of concrete can be calculated by equation 19.2.3.1 for normal weight aggregate as:

𝑓 7.5 𝑓 𝑐 𝑓 375 𝑝𝑠𝑖

Accordingly, the cracking strain of concrete can be estimated as:

𝜀 𝑓 𝐸

7.5 𝑓 𝑐

57000 𝑓 𝑐 0.00013

Similarly, the cracking moment and maximum moment at the middle of the reinforced concrete slab under service load can be calculated with reference to ACI table 6.5.2 as:

𝑀 𝑓 𝐼 𝑐

1.266 𝑘𝑖𝑝. 𝑓𝑡

𝑀 𝑤 𝑤 𝑙 𝑤 𝑙

2.255 𝑘𝑖𝑝 𝑓𝑡

Since the service load moment (𝑀 ) is greater than the cracking moment (𝑀 ) the section is observed to be cracked under service loads. To inspect the locations where the slab is likely to be cracked based on the exerted stress, the finite element model was used incorporating the dead loads and the time-dependent effects of creep and shrinkage. The finite element model presenting the total strains under service loads including time-dependent effects is shown in Figure 5. The figure confirms the simplified analysis using ACI methods that under service loads and time-dependent effects, the reinforced concrete slab is likely to be cracked at multiple locations.

Figure 5: total strain under service loads (dead + live + reduced superimposed) where colors in blue show strains exceeding the cracking strain of concrete. Areas shown in blue are likely to be cracked under service loads.

As a result, a stiffness reduction factor was calculated for both 4.5 in and 5 in slabs following Branson’s (1977) method shown in Figure 12 and attached in the appendix. As a result, the slab was treated as partially or fully cracked per the strain map as shown in Figure 6. A schematic showing the cracking reduction factor in the slab based on the finite element model is shown in Figure 13.

Figure 12: Moment of inertia effects of partially cracked (noted herein as cracked) versus fully cracked section.

Figure 6: A schematic showing slab areas where stiffness reduction was used per the calculations provided in the appendix.

8. Non-destructive testing of the building slab

A non-destructive field-testing program was conducted by a third-party subcontractor. The non-destructive testing included taking four core samples and one sample of steel rebar. The location of the samples extracted from the first and second floors are shown in Figure 14 and Figure 15 respectively. A few snapshots from the core marking, coring process, the process to extract the steel rebar, and an example core sample are shown in Figure 16, Figure 17, Figure 18, and Figure 19 respectively.

Figure 7: A schematic showing the core locations taken on the first floor of the building.

Figure 8: A schematic showing the core locations and the rebar sample location taken on the second floor of the building.

Figure 16: Example mapping of the core and rebar extraction locations on the slabs.

Figure 9: Coring of the slab to extract concrete samples

Figure 10: Extraction of steel rebar from the top of the slab.

Figure 11: Example concrete core extracted from the slab.

8.1.Non-destructive testing results

The laboratory observations of the cores extracted from the existing slab are presented in Error!

Not a valid bookmark self-reference.. To estimate the compressive strength based on the non-destructive testing, the ACI 214.4R-03 (Guide for obtaining cores and interpreting compressive strength results) was used to calculate an effective design compressive strength. The alternative method described by the statistical equation below was used to estimate the design strength of concrete

Table 5: Testing results of drilled cores of concrete per ASTM C42-20

Core location Length

Rec. (in)

Length before cap

(in)

Avg Diameter

(in)

Compressive strength Uncorrec ted (psi)

Correctio n factor

Correct ed (psi)

6.5' N. of S. Wall, 6.5' W. of E. Wall

3.7 3 1.76 3,770 1 3,770

11.5' N. of S. Wall 6.5' W. of E. Wall

5.6 3.2 1.78 3,460 1 3,460

8' N. of S. Wall, 16.5' W. of E. Wall

3.5 3.4 1.77 4,670 1 4,670

7' N. of S. Wall, 7.5' W. of E. Wall

4.7 2.6 1.76 5,150 0.9633 4,960 𝑓 𝐶 𝑓 𝑇 𝑆 𝑛

𝑍 𝑆

Table 6 presents the concrete design strength based on the statistical analysis methods provided by ACI 214.4R-03. As it can be observed, the design compressive strength is 2804 psi at 90% confidence interval and 2569 psi at 95% confidence interval. Using a 95% confidence interval for conservative predictions, it is apparent that design compressive strength is 2569 psi which is 2.8% different than the 2500 psi design strength suggested on the drawings and used for the above analysis. It is fair to assume that the compressive strength of the slab is 2500 psi.

Table 6: Compressive strength estimate based on non-destructive testing.

Confidence interval Design compressive strength (psi) 90% confidence level 2804 95% confidence level 2569

8.2.Updated model and calculations based on the results of field testing.

Since the difference between the field-tested concrete design strength and the concrete design strength listed on the as-built drawings is less than 5%, which is the typical error range of the finite element model, no update of the finite element model is necessary. The original design compressive strength and model shown above are valid and shall be good to predict the deflections of the floor slabs.

9. Deflection prediction

The finite element model deflection prediction of the second-floor slab incorporating the effects of cracking and time-dependent effects of creep and shrinkage is shown in Figure .

Figure 20: Second floor deflection map considering reduced stiffness due to cracking as well time-dependent effects of creep and shrinkage while subjected to self-weight and reduced super imposed dead loads.

The results show significantly large deformations reaching 2.7 in under only self-weight and the reduced superimposed loads. The model predicted deflections are compared to the field measurements as presented in Table . The model captures the deflections exhibited by the structure very well. All prediction errors are within ±10% proving the finite element model fidelity. This in turn validates the previous calculations of cracking and time-dependent effects.

Table 7: Comparing deflections predicted using the finite element model with these measured in the field. All prediction errors are within ±10% proving the finite element model fidelity.

Point A B C D E F G Inspection (in) 2.0 2.5 2.0 3.0 1.5 2.0 1.8 Model (in) 2.1 2.6 2.0 2.7 1.6 2.1 1.9 % difference 8% 2% 0% -10% 4% 8% 6%

The above analysis shows that the slab design is governed by deflection. Unless the deflection due to self-weight and reduced superimposed dead load can be reduced, the slab will not meet the ACI code requirements under any live load. The current live load capacity of the slab based on serviceability requirements by the ACI code is zero.

10. Allowable live load capacity

Per the structural drawings, the slab moment capacity was evaluated with reference to ACI 318- 14 for both service and ultimate states. The following described parameters were used to determine the original permissible live loads of the original slab:

Reinforcement diameter = 0.5 in Reinforcement spacing at the bottom of the slab = 8 in Reinforcement spacing at the top of the slab = 12 in Reinforcement cover = 0.75 in

The design capacity based on ultimate limit state was calculated for the concrete and steel contribution using the following equations:

𝑎 𝜙𝑀 𝐴 𝑓 𝑑 𝑎

3.5 𝑘𝑖𝑝.𝑓𝑡

Based on the available moment capacity, the allowed service live loads are:

𝑤 _ 10 𝜙𝑀 𝑙 𝑤 𝑤 102.5 𝑝𝑠𝑓

The allowed live load due at ultimate load capacity is:

𝑤 _

52 𝑝𝑠𝑓 conservatively 50 psf

The original slab design allows for a maximum of 50 psf for live load at the reduced superimposed dead load which is greater than the described design loads of 40 psf.

Given the above analysis for ultimate limit state and serviceability limit state capacities based on the ACI code requirements, the live load capacity of the as-built slab is zero.

11. Lateral diaphragm capacity

The lateral transfer diaphragm capacity of the as-built slab per both UBC-1964 and IBC-2018 and detailed calculations are provided in the Appendix. For lateral loading, analysis was performed for both the complete structure tied by the foundation as well as two separate structures identified as center and secondary structures as shown in Figure . Analysis for the lateral transfer capacity checked both the total shear exerted on the structure versus the slab capacity as well as identifying the rigid slab section acting as diaphragms as shown in Figure .

Finally, because the East-West (E-W) direction is significantly longer than the North-South (N-S) direction as well as containing all beams parallel to that direction, the lateral transfer capacity check was completed for the N-S direction as the critical loading direction.

Figure 21: Identification of the center and secondary structures as the two main parts of the complete structure for lateral load analysis.

Figure 22: Identification of the locations of the main diaphragms in comparison to the complete slab diaphragm.

11.1. Lateral transfer capacity of the original slab diaphragm per UBC 61-64

The following are the key parameters used in this analysis:

Numerical coefficient of seismic zone (Z) = 2, per UBC-1964 maps Horizontal force factor (K) = 1, per UBC-1964 Table No 23.H Critical lateral load direction = N-S direction Wind pressure = 30 psf Horizontal force factor (Cp) = 0.1

The fundamental period of vibration of the N-S was calculated as:

𝑇

0.05 ℎ

√𝐷

The base shear was calculated using:

𝑉 𝑍 𝐾 𝐶 𝑊

Then distributed in the height of the structure using:

𝐹 0.004𝑉 ℎ 𝐷

𝐹 𝑉 𝐹 𝑤 ℎ ∑ 𝑤 ℎ

Analysis was completed for the induced shear with either seismic or wind loads. However, because the seismic loads were more critical than wind loads, the diaphragm load transfer capacity was evaluated using seismic loads. The lateral forces for seismic analysis of the original diaphragm are presented in Table 8.

Table 8: Lateral forces for seismic analysis of the original diaphragm

Value Complete structure

Center structure

Secondary structure

Fundamental period of vibration (sec) 0.856 0.648 0.517 Base shear (kip) 807 313 404 Force on roof (kip/ft) 0.721 0.525 - Force on 3rd floor (kip/ft) 0.707 0.425 0.698 Force on 2nd floor (kip/ft) 0.439 0.232 0.413

The factored shear capacity of concrete for 2,500 psi compressive strength is:

∅𝑉 0.75 100 𝑡 𝑑

The reinforcement shear capacity is given by:

∅𝑉 ∅𝐴 𝑓 𝑑 𝑠

The total shear capacity for the main diaphragm is given by:

∅𝑉 ∅𝑉 ∅𝑉

For a conservative assumption and due to the heavy cracking of the slab, we assumed that the total slab diaphragm (shear transfer) capacity is governed by concrete only without considering steel reinforcement. Therefore, the slab diaphragm capacity is calculated as:

∅𝑉 ∅𝑉

The slab diaphragm load and capacity are compared in Table 9.

Table 9: Seismic loads and diaphragm capacity of the original slab diaphragm per UBC1964.

Shear (kip) Center structure

Secondary structure

Complete structure

Slab diaphragm Load 138 241 380 Capacity 149 261 410

The tension generated by the lateral forces due to the bending moment was calculated using:

𝑇 𝑀

This requires collector reinforcement as given by:

𝐴

0.9 𝑓

Table 10: Collector reinforcement of the main diaphragm.

Collector information Center structure

Secondary structure

Complete structure

Beam designation B405 B304 B405 Top reinforcement information

Number of rebars 2 7 2 Bar diameter (in) ¾ 5/8 ¾

Capacity (in2) 0.88 2.14 0.88 Demand (in2) 0.07 0.07 0.10

The reinforcement of the collectors in the main diaphragm were checked for the resulting bending moments and the top reinforcement in the existing beams were found to exceed the steel requirement demand as presented in Table 10 and as shown in the Appendix.

The above analysis shows that the original concrete slab can safely act as a diaphragm to transfer the lateral forces on the building as per UBC 1964 requirements.

11.2. Lateral transfer capacity of the as-built slab diaphragm per IBC 2018

The following are the key parameters used in this analysis:

The design spectral response acceleration parameter in the short period range, SDS = 0.38 The design spectral response acceleration parameter at period of 1.0 s, SD1 = 0.22 The mapped maximum considered earthquake spectral response acceleration, S1 = 0.12 The response modification factor, R = 5 Risk category = I Importance factor, Ie = 1

Similar check was completed of the lateral transfer capacity of the original slab per IBC 2018 to the one performed in the previous step where the fundamental period was approximated by:

𝑇 𝐶 ℎ

The vertical distribution factor is given by:

𝐶 𝑤 ℎ

∑ 𝑤 ℎ

The base shear is given by:

𝐶 𝑆 𝑅 𝐼

𝑉 𝐶 𝑊

The diaphragm design forces are calculated using:

𝐹 0.2𝑆 𝐼 𝑤 ∑ 𝐹 ∑ 𝑤 𝑤 0.4𝑆 𝐼 𝑤

Table 1 1: Lateral forces for seismic analysis of the original diaphragm

Value Complete structure Center structure Secondary structure Fundamental period of vibration (sec)

0.342

Base shear (kip) 492 266 226 Force on roof (kip/ft) 0.44 0.38 - Force on 3rd floor (kip/ft) 0.49 0.37 0.46 Force on 2nd floor (kip/ft) 0.49 0.37 0.41

The factored shear capacity of concrete based on 2,500 psi compressive strength is:

∅𝑉 ∅ 2 𝑓 𝑡 𝑑

The reinforcement shear capacity is given by:

∅𝑉 ∅𝐴 𝑓 𝑑 𝑠

The total shear capacity for the main diaphragm is given by:

∅𝑉 ∅𝑉 ∅𝑉

For a conservative assumption and due to the heavy cracking of the slab, we assumed that the total slab diaphragm (shear transfer) capacity is governed by concrete only without considering steel reinforcement. Therefore, the slab diaphragm capacity is calculated as:

∅𝑉 ∅𝑉

Table 12: Seismic loads per IBC 2018 of the original slab diaphragm

Shear (kip) Center structure Secondary structure

Complete structure

Slab diaphragm Load 100 159 261 Capacity 149 261 410

The tension generated by the lateral forces due to the bending moment were calculated using:

𝑀

This requires collector reinforcement as given by:

𝐴

0.9 𝑓

Table 13: Collector reinforcement of the main diaphragm.

Collector information Center structure

Secondary structure

Complete structure

Beam designation B405 B204 B204 Top reinforcement information

Number of rebars 2 2 2 Bar diameter (in) ¾ ¾ ¾

Capacity (in2) 0.88 0.88 0.88 Demand (in2) 0.06 0.12 0.08

The reinforcement of the collectors in the main diaphragm were checked for the resulting bending moments and the top reinforcement in the existing beams was found to exceed the steel requirement demand as presented in Table 13 and as presented in the Appendix.

Similar to the check of the UBC 1964, the above analysis using INC 2018 shows that the original concrete slab can safely act as a diaphragm to transfer the lateral forces on the building as per IBC 2018requirements.

12. Repair/Strengthening Strategy

The RC slab in the building does not meet the serviceability requirements as its deflection exceed the deflection limits by defined by ACI 318-14. To repair/strengthening the concrete slabs, we suggest the use of an external unbonded post-tensioning system. The use of external post-tensioning has been demonstrated in the literature as an effective method to restore the deflection of the concrete floor due to sustained and live loads. The external post-tensioning offers several advantages over other repair methods. These advantages include the followings:

a. Minimum added weight to the concrete slab.

b. No shoring of the structural element is required.

c. Limited construction activities and construction cost (limited drilling).

d. Rapid repair with limited time required for the installation of post-tensioning system compared with long time required for conventional methods of repair.

e. Adequate restoration for both deflection control (service limit state) and slab capacity

(ultimate limit state)

f. Control/close of existing cracks.

g. The system will be enclosed in the false ceiling so it will not be observed by the users.

12.1. External Post-tensioning Design

The design of external post-tensioning system for slab repair or strengthening requires the determination of the profile of the pot-tensioning strands, the diameter and spacing of the strands, and the level of post-tensioning (PT) forces. Figure 23 provides examples of some profile of PT that can be used to restore the concrete slab capacity. PT strands may be applied for single end spans with one-point or two-point draped profiles. PT strands may also be continuous through the three spans. The edge and intermediate beams can be utilized to support the hydraulic jacks and anchors required for stressing the PT strands. The steel king posts are utilized to transmit the PT forces at the draping points.

Figure 23: Plan of the structural slab to be post-tensioned. Section A-A is shown in Figure 24 to demonstrate example post-tensioning systems that can be used to restore slab capacity.

Figure 24: Examples of external post-tensioning profiles for concrete slab that can be used to restore the RC slab capacity

The eccentricities for the strands shall be carefully designed to provide proper moments and deflections that counteract the applied gravity loads. Figure 25 shows schematic representation for the moments due to PT and the corresponding deflected shape of the concrete slab due to single-span one-point draped PT strands. In addition to moment effects, the axial forces due to PT help restore the rigidity of the concrete floor by closing the existing cracks.

Section A‐A

16 ft16 ft 8.5 ft

Continuous strands

Single‐span strands One drape point

Two drape points Bolts

False ceiling

False ceiling Steel post

PT strand

Concrete floor

Base plate

Figure 25: Schematic for the effect of PT on moments and deflection of concrete floor

12.2. Evaluation of selected repair strategy

The goal of evaluating the repair strategy is to determine how efficiently utilizing lateral post-tension can control excessive deflection and restore the serviceability of the concrete floors. To perform the evaluation, PT strands were created in the ETABS model in the one-floor panel that belongs to the second floor where the deflections were field measured. The location of the floor panel and the profile details of the added PT strands are shown in Figure 26 and Figure 27, respectively. In this evaluation, four PT strands were used within the panel at a spacing of 4.5 ft.

Figure 26: Location of PT strands spaced at 4.5 ft

16 ft16 ft 8.5 ft e1 e2

Primary Moment due to PT

Deflection due to PT (counteracting gravity loads)

Upward deflection

Maximize eccentricity minimize eccentricity minimize eccentricity

Maximize eccentricity minimize eccentricity minimize eccentricity

14”

13”

18.5”

13”

42” 42” 15”

Secondary Moment due to PT

PT

Effect

Final Moment due to

PT

Figure 27: PT strands created in the ETABS model to simulate external PT system.

12.3. Results of deflection analysis using post-tensioning

Figure 28 shows the contour for the deflection of the cracked concrete slab due to dead and half the superimposed dead loads. The deflection due to these sustained loads is 2.5 inches, which exceeds the ACI deflection limit due to sustained loads (Lmax/240 = 0.75 inch). Figure 29 shows that application of PT results in -3.1 inch upward deflection. The net deflection due to sustained load and PT forces is -0.6 inch upward for the floor panel. The deflection due to live loads is found to be 0.85 inch as shown in Figure 30. The deflection due to full-service loads is estimated to be 0.25 inch. A summary for deflection due to various loads is listed in Table 14.

Figure 28: Deflection contours due to sustained loads in inches

Figure 29: Deflection contours due to PT forces in inches

Figure 30: Deflection contours due to live loads in inches

Table 14: Deflections of concrete floor due to service loads and PT

Loads Value Cumulative

Dead + Superimposed Dead (cracked + long term) 2.5 in. 2.5 in.

Post-tensioning -3.1 in. -0.6 in.

Live Load 0.85 in. 0.25 in.

The final deflection for the floor panel meets the ACI deflection limits for deflection due to total service loads (Lmax/240 = 0.75 inch) as well as deflection due to live loads only (LLL/360 = 0.6 inch). Based on deflection analysis, the allowable live load can increase by 25% from 40 psf to 50 psf.

12.4. Ultimate capacity of the post-tensioned slab

Figure 31: Floor mid-span strain distribution and force equilibrium.

Figure 31 shows floor mid-span strain distribution and force equilibrium. Based on force equilibrium, the depth of the neutral axis and nominal moment capacity was calculated in the following equations. The nominal moment capacity was computed as 12.1 kip-ft, which is roughly three times the nominal moment capacity of concrete floor without PT.

𝑎

𝑀 𝐴 𝑓 𝑑 𝐴 𝑓 𝑑

The allowable live loads due to the use of PT was also calculated following two equations.

𝑀 . . 𝑀 3.05 𝑘𝑖𝑝.𝑓𝑡 𝑤 _ 𝑤 𝑤 250 𝑝𝑠𝑓

Following the increase in moment capacity, the allowable live load also increases from 50 psf to 250 psf. The analysis results indicate that the governing limit state in the design of the concrete floor is the service limit state, and the deflection limits govern the determination of allowable live loads. The ultimate load capacity does not govern the slab capacity before or after post-tensioning.

Strain Distribution Force Equilibrium

13. Conclusions and Recommendations

Analysis using the finite element method of the as built reinforced concrete slab in AFB building 20202 showed the concrete slab not to meet code requirements for serviceability due to excessive deflection under self-weight and reduced superimposed live load. The situation becomes worse under the live load. All assessments showed the slab to be mostly cracked. Non-destructive testing was performed and showed the concrete to have a strength of 2500 psi similar to that listed on the as-built drawings. Therefore, no update of the finite element model was necessary.

Analysis shows that the live load capacity of the slab based on ultimate limit state is 50 psf.

However, with the slab not meeting serviceability limit state requirements by the ACI under dead loads, the allowable live load capacity of the as-built slab is zero. Analysis also showed the slab meets ultimate limit state requirements for lateral load transfer under both the UBC-61 1964 and the current IBC 2018 codes. It is evident that the slab can safely transfer the lateral loads as a diaphragm.

Repair of the reinforced concrete slab to restore flexural capacity and be serviceable is suggested using the lateral post-tensioning technique. A profile is suggested to be used and analysis for the concrete floor post-tensioned laterally was performed and presented. The post-tensioning method proved able to eliminate the deflection problem and thus to restore the slab capacity. The proposed repair method also proved able to increase moment capacity almost three times the moment capacity of the concrete floor without PT. The allowable live load is governed by the ACI deflection limit due to live loads, which can be increased PT is used. An allowable live load of 50 psf can be used if the slab is laterally post-tensioned.

While the analysis proves the effectiveness of PT in restoring floor deflection and increasing floor capacity, detailed analysis of the PT system will be required to consider the following factors in the repair design.

• Optimum PT profile to achieve target live loads with minimum construction cost.

• Effect of secondary moments due to PT.

• Effect of PT on negative moment and restrained regions near columns (to avoid cracking).

• Varying PT spacing throughout the building.

• Check shear capacity to resist higher loads.

14. Appendix

Elastic analysis of cracked flexural member

Compressive Strength ≔f'c 2.5 ksi

≔Es 29000 ksi

≔Ec =⋅57000 ‾‾‾‾‾‾‾⋅f'c 1 psi ⎛⎝ ⋅2.85 103 ⎞⎠ ksi

≔n ― Es

Ec

Dimensions

≔b 12 in ≔h 4.5 in ≔c ― h spacing

≔s1 8 in ≔s2 16 in

≔As1 ⋅⋅π ――― in b s1

≔As2 ⋅⋅π ――― in b s2

Reinforcement ratio

≔ρ1 ―― As1

⋅b h ≔ρ2 ――

As2

⋅b h

≔k1 -‾‾‾‾‾‾‾‾‾‾‾‾‾+⋅2 ρ1 n ⎛⎝ ⋅ρ1 n⎞⎠ 2 ⋅ρ1 n

≔k2 -‾‾‾‾‾‾‾‾‾‾‾‾‾+⋅2 ρ2 n ⎛⎝ ⋅ρ2 n⎞⎠ 2 ⋅ρ2 n

≔j1 -1 ― k1

≔j2 -1 ― k2

Non-Commercial Use Only

Taha Rectangle

Service limit state

≔DL ⋅⋅4.5 in 150 pcf 1 ft ≔LL 40 psf ≔SDL 8 psf

≔lslab1 14.75 ft

≔lslab2 7.333 ft

≔MSLS1

⋅(( +DL ⋅⋅0.5 SDL 1 ft)) lslab1

⋅(( ⋅LL 1 ft)) lslab1

=MSLS1 2.181 ⋅kip ft

≔MSLS2

⋅(( +DL ⋅SDL 1 ft)) lslab2

⋅(( ⋅LL 1 ft)) lslab2

≔fs1 ―――

MSLS1

⋅⋅As1 j1 h ≔fs2 ―――

MSLS2

⋅⋅As2 j2 h

Uncracked section

≔Ig ⋅b ― h3

≔yt ― h

Assuming 0.75 in steel cover

≔It1 +Ig ⋅As1 (( -n 1)) ⎛⎝ -yt 0.75 in⎞⎠

≔It2 +Ig ⋅As2 (( -n 1)) ⎛⎝ -yt 0.75 in⎞⎠

Non-Commercial Use Only

≔It2 +Ig ⋅As2 (( -n 1)) ⎛⎝ -yt 0.75 in⎞⎠

For normal weight concrete ≔λ 1

≔fr ⋅7.5 λ ‾‾‾‾‾‾‾⋅1 psi f'c

Cracking moment capacity

≔Mcr ―― ⋅fr Ig yt

=Mcr 1.266 ⋅kip ft

≔Check1 if ⎛⎝ ,,≤Mcr MSLS1 “Slab is cracked” “Uncracked slab”⎞⎠

≔Check2 if ⎛⎝ ,,≤Mcr MSLS2 “Slab is cracked” “Uncracked slab”⎞⎠

=Check1 “Slab is cracked” =Check2 “Uncracked slab”

≔Icr1 +―――― ⋅b ⎛⎝ ⋅k1 h⎞⎠

⋅⋅⋅n As1 ⎛⎝ -1 k1⎞⎠

2 h2 =Icr1 39.46 in 4

≔Icr2 +―――― ⋅b ⎛⎝ ⋅k2 h⎞⎠

⋅⋅⋅n As2 ⎛⎝ -1 k2⎞⎠

2 h2 =Icr2 22.313 in 4

Non-Commercial Use Only

5" slab

≔h3 5 in

≔Ig3 ⋅b ―― h3 spacing

≔s3 7 in

≔As3 ⋅⋅π ――― in b s3

≔ρ3 ―― As3

⋅b h3

≔k3 -‾‾‾‾‾‾‾‾‾‾‾‾‾+⋅2 ρ3 n ⎛⎝ ⋅ρ3 n⎞⎠ 2 ⋅ρ3 n

≔lslab3 14.75 ft

≔MSLS3

⋅(( +DL ⋅SDL 1 ft)) lslab3

⋅(( ⋅LL 1 ft)) lslab3

≔j3 -1 ― k3

≔yt3 ― h3

≔It3 +Ig3 ⋅As3 (( -n 1)) ⎛⎝ -yt3 0.75 in⎞⎠

≔Icr3 +―――― ⋅b ⎛⎝ ⋅k3 h3⎞⎠

⋅⋅⋅n As3 ⎛⎝ -1 k3⎞⎠

2 h3 2 =Icr3 55.349 in 4

≔Mcr3 ―― ⋅fr Ig3 yt3

=Mcr3 1.563 ⋅kip ft

≔Check3 if ⎛⎝ ,,≤Mcr3 MSLS3 “Slab is cracked” “Uncracked slab”⎞⎠

=Check3 “Slab is cracked”

Non-Commercial Use Only

Stiffness reduction factors

≔Ie1 min

,+Icr1 ⋅⎛⎝ -Ig Icr1⎞⎠ Mcr

MSLS1

It1

=Ie1 49.555 in 4

≔Ie2 min

,+Icr2 ⋅⎛⎝ -Ig Icr2⎞⎠ Mcr

MSLS2

It2

=Ie2 94.165 in 4

≔Ie3 min

,+Icr3 ⋅⎛⎝ -Ig3 Icr3⎞⎠ Mcr3

MSLS3

It3

=Ie3 78.121 in 4

Stiffness factors if partially cracked

≔η1 if

,,≤Mcr MSLS1 ―― Ie1 It1

=η1 0.51

≔η2 if

,,≤Mcr MSLS2 ―― Ie2 It2

=η2 1

≔η3 if

,,≤Mcr3 MSLS3 ―― Ie3 It3

=η3 0.581

Stiffness factors if fully cracked

≔ηf1 if

,,≤Mcr MSLS1 ―― Icr1 It1

=ηf1 0.406

≔ηf2 if

,,≤Mcr MSLS2 ―― Icr2 It2

=ηf2 1

≔ηf3 if

,,≤Mcr3 MSLS3 ―― Icr3 It3

=ηf3 0.412

Non-Commercial Use Only

Creep coefficient calculations based on CEB-FIP Model Code 90

Materials properties and creep parameters ≔f'c 2.5 ksi

≔Ec =⋅57000 ‾‾‾‾‾‾‾⋅f'c 1 psi ⎛⎝ ⋅2.85 103 ⎞⎠ ksi

Slab dimensions ≔bs 1 ft ≔hs 4.5 in

Area of the concrete member

≔Ac ⋅bs hs

Modulus of elasticity at 28 days

≔fcm f'c ≔fck +f'c 8 MPa ≔fcm0 10 MPa code constant

National size of member, hc

Perimeter of member in contact with atmosphere, u

≔u ⋅2 ⎛⎝bs⎞⎠

≔hc ⋅2 ― Ac u ≔h0 100 mm code constant

Relative Humidity

≔RH %50 ≔RH0 %100 code constant

≔βH +⋅⋅150

+1

1.2 RH

RH0

18⎞ hc h0

250 =βH 421.467

≔ϕRH +1 ――――

-1 ――

RH

RH0

0.46 ‾‾‾3 hc h0

=ϕRH 2.04

≔βfcm ――― 5.3 fcm fcm0

=βfcm 4.037

Non-Commercial Use Only

≔t0 7 day Number of days when load is applied

≔t1 1 day code constant

≔βt0 ―――――

+0.1 t0 t1

0.2⎞

=βt0 0.635

Age of structure when creep deflection is calculated

≔t ⋅59 365 day

≔tt_t0 -t t0

Creep coefficient

≔βc ((t)) ――――

⎛⎝ -t t0⎞⎠ t1

+βH ――― ⎛⎝ -t t0⎞⎠ t1

0.3

=βc ((t)) 0.994[[ ]]

≔βcc e

⋅0.25

-1 t t1

0.5⎞

⎠ =βcc 1.273

≔ϕ0 ⋅⋅ϕRH βt0 βfcm =ϕ0 5.225

≔Eci ⋅‾‾‾βcc Ec

≔C ⋅ϕ0 βc ((t)) =C 5.195[[ ]]

≔εcr ⋅C ε0

≔Ecr ――― Eci

(( +1 C)) =Ecr 518.974[[ ]] ksi

Non-Commercial Use Only

Shrinkage according to CEB/FIP Model code 90

≔βsc 5 For normal hardening cement

≔ts 1 day Age of concrete at beginning of shrinkage

≔εsfcm ⋅+160 ⋅⋅10 βsc

-9 ―― fcm fcm0

10-6

≔βRH ⋅-1.55 -1

RH

RH0

3⎡

≔εcs0 ⋅εsfcm βRH

≔tt_ts -t ts

≔βstt_ts

-t ts t1

+350 hc h0

2 ⎛ -t ts t1

≔εcs ⋅-εcs0 βstt_ts =εcs ⋅7.03 10-4

From ETABS model ≔ε0 0.00007 ≔σDL_0 191 psi

≔εcr ⋅C ε0 =εcr ⋅3.636 10-4⎡⎣ ⎤⎦

≔εT ++ε0 ⋅C ε0 εcs

≔Eeff ―― σDL_0 εT =Eeff 168.04[[ ]] ksi

Non-Commercial Use Only

Floor weights, complete structure:

≔Wroof (( ++456.5 336.3 ⋅96 0.5)) kip ≔hroof 30 ft

≔W3rd (( +++1235.5 1046.6 ⋅273 0.5 96.5)) kip ≔h3rd 20 ft

≔W2nd (( +++++1476 1327 9 5 ⋅333 0.5 ⋅273 0.5)) kip ≔h2nd 10 ft

≔W1st (( +++1351 1522 ⋅219 0.5 ⋅333 0.5)) kip ≔h1st 0 ft

Floor weights, center structure Floor weights, secondary structure

≔Wroof_c

++456.5 336.3 ― kip ≔Wroof_s -Wroof Wroof_c

≔W3rd_c

++462 452 ――― +120 96 kip ≔W3rd_s -W3rd W3rd_c

≔W2nd_c

++513 462 ―――― +157 120 kip ≔W2nd_s -W2nd W2nd_c

≔W1st_c

++541 796 ―――― +157 157 kip ≔W1st_s -W1st W1st_c

Total weight of floors

≔W ++Wroof W3rd W2nd

≔Wc ++Wroof_c W3rd_c W2nd_c

≔Ws ++Wroof_s W3rd_s W2nd_s

Shear capacity according to ASCE 7-16

Non-Commercial Use Only

≔Ws ++Wroof_s W3rd_s W2nd_s

Shear capacity according to ASCE 7-16

N-S center structure parallel dimension ≔DNS_c 442 in

N-S Secondary structure parallel dimension ≔DNS_s 772 in

N-S Complete structure ≔DNS +DNS_c DNS_s

≔SMS 0.58

≔SDS 0.38

≔Ss 0.43

≔SD1 0.22

≔R 5

≔Cd 4.5

≔Cu 1.48

≔Ie 1

≔Cs ――

SDS

R Ie

Non-Commercial Use Only

≔check_1 if ⎛⎝ ,,>Cs ⋅⋅0.044 SDS Ie “Good” “Not good”⎞⎠

=check_1 “Good”

≔V =⋅Cs W 492.168 kip

Approximate fundamental period (12.8-7)

≔Ct 0.016 sec ≔x 0.9 ≔hn 30

≔Ta ⋅Ct hn x =Ta 0.342 s ≔k 1

Base shear

Center structure ≔Vc =⋅Cs Wc 226.199 kip

Secondary structure ≔Vs =⋅Cs Ws 265.97 kip

Complete structure

For complete structure

≔sum_1 ⎛⎝ ++⋅Wroof hroof k ⋅W3rd h3rd k ⋅W2nd h2nd k ⎞⎠

≔Cv_roof ―――― ⋅Wroof hroof k sum_1 ≔Cv_3rd ――――

⋅W3rd h3rd k sum_1 ≔Cv_2nd ――――

⋅W2nd h2nd k sum_1

≔Froof ⋅Cv_roof V ≔F3rd ⋅Cv_3rd V ≔F2nd ⋅Cv_2nd V

=Froof 116.321 kip =F3rd 231.968 kip =F2nd 143.879 kip

≔Vroof Froof ≔V3rd +F3rd Froof ≔V2nd ++F2nd F3rd Froof

Diaphragm design forces

≔Fp_roof_1 ⋅―― Vroof

Wroof

Wroof

≔Fp_3rd_1 ⋅―――― V3rd

+Wroof W3rd

W3rd

≔Fp_2nd_1 ⋅――――――― V2nd

++Wroof W3rd W2nd

W2nd

Non-Commercial Use Only

≔Fp_roof_2 ⋅⋅⋅0.2 SDS Ie Wroof ≔Fp_roof_3 ⋅⋅⋅0.4 SDS Ie Wroof

≔Fp_3rd_2 ⋅⋅⋅0.2 SDS Ie W3rd ≔Fp_3rd_3 ⋅⋅⋅0.4 SDS Ie W3rd

≔Fp_2nd_2 ⋅⋅⋅0.2 SDS Ie W2nd ≔Fp_2nd_3 ⋅⋅⋅0.4 SDS Ie W2nd

Diaphragm design forces

≔Fp_roof min ⎛⎝ ,max ⎛⎝ ,Fp_roof_2 Fp_roof_1⎞⎠ Fp_roof_3⎞⎠ =Fp_roof 116.321 kip

≔Fp_3rd min ⎛⎝ ,max ⎛⎝ ,Fp_3rd_2 Fp_3rd_1⎞⎠ Fp_3rd_3⎞⎠ =Fp_3rd 261.027 kip

≔Fp_2nd min ⎛⎝ ,max ⎛⎝ ,Fp_2nd_2 Fp_3rd_1⎞⎠ Fp_2nd_3⎞⎠ =Fp_2nd 261.027 kip

Floor shear forces based on N-S direction

≔Fp_roof_NS ――― Fp_roof

264.5 ft =Fp_roof_NS 0.44 ―― kip ft

≔Fp_3rd_NS ――― Fp_3rd

537.5 ft =Fp_3rd_NS 0.486 ―― kip ft

≔Fp_2nd_NS ――― Fp_2nd

537.5 ft =Fp_2nd_NS 0.486 ―― kip ft

For center structure

≔sum_2 ⎛⎝ ++⋅Wroof_c hroof k ⋅W3rd_c h3rd k ⋅W2nd_c h2nd k ⎞⎠

≔Cv_roof_c ――――― ⋅Wroof_c hroof k sum_2 ≔Cv_3rd_c ―――――

⋅W3rd_c h3rd k sum_2 ≔Cv_2nd_c ―――――

⋅W2nd_c h2nd k sum_2

≔Froof_c ⋅Cv_roof_c Vc ≔F3rd_c ⋅Cv_3rd_c Vc ≔F2nd_c ⋅Cv_2nd_c Vc

=Froof_c 100.453 kip =F3rd_c 81.401 kip =F2nd_c 44.345 kip

≔Vroof_c Froof_c ≔V3rd_c +F3rd_c Froof_c ≔V2nd_c ++F2nd_c F3rd_c Froof_c

Non-Commercial Use Only

Diaphragm design forces

≔Fp_roof_c_1 ⋅――― Vroof_c

Wroof_c

Wroof_c

≔Fp_3rd_c_1 ⋅――――― V3rd_c

+Wroof_c W3rd_c

W3rd_c

≔Fp_2nd_c_1 ⋅―――――――― V2nd_c

++Wroof_c W3rd_c W2nd_c

W2nd_c

≔Fp_roof_c_2 ⋅⋅⋅0.2 SDS Ie Wroof_c ≔Fp_roof_c_3 ⋅⋅⋅0.4 SDS Ie Wroof_c

≔Fp_3rd_c_2 ⋅⋅⋅0.2 SDS Ie W3rd_c ≔Fp_3rd_c_3 ⋅⋅⋅0.4 SDS Ie W3rd_c

≔Fp_2nd_c_2 ⋅⋅⋅0.2 SDS Ie W2nd_c ≔Fp_2nd_c_3 ⋅⋅⋅0.4 SDS Ie W2nd_c

≔Fp_roof_c min ⎛⎝ ,max ⎛⎝ ,Fp_roof_c_2 Fp_roof_c_1⎞⎠ Fp_roof_c_3⎞⎠ =Fp_roof_c 100.453 kip

≔Fp_3rd_c min ⎛⎝ ,max ⎛⎝ ,Fp_3rd_c_2 Fp_3rd_c_1⎞⎠ Fp_3rd_c_3⎞⎠ =Fp_3rd_c 99.772 kip

≔Fp_2nd_c min ⎛⎝ ,max ⎛⎝ ,Fp_2nd_c_2 Fp_3rd_c_1⎞⎠ Fp_2nd_c_3⎞⎠ =Fp_2nd_c 99.772 kip

Floor shear forces based on N-S direction

≔Fp_roof_c_NS ――― Fp_roof_c

264.5 ft =Fp_roof_c_NS 0.38 ―― kip ft

≔Fp_3rd_c_NS ――― Fp_3rd_c

264.5 ft =Fp_3rd_c_NS 0.377 ―― kip ft

≔Fp_2nd_c_NS ――― Fp_2nd_c

264.5 ft =Fp_2nd_c_NS 0.377 ―― kip ft

Non-Commercial Use Only

For secondary structure

≔sum_3 ⎛⎝ ++⋅Wroof_s hroof k ⋅W3rd_s h3rd k ⋅W2nd_s h2nd k ⎞⎠

Diaphragm design forces

≔Cv_3rd_s ――――― ⋅W3rd_s h3rd k sum_3 ≔Cv_2nd_s ―――――

⋅W2nd_s h2nd k sum_3

≔Fp_3rd_s_1 ⋅――― V3rd_s

W3rd_s

W3rd_s

≔F3rd_s ⋅Cv_3rd_s Vs ≔F2nd_s ⋅Cv_2nd_s Vs

≔Fp_2nd_s_1 ⋅――――― V2nd_s

+W3rd_s W2nd_s

W2nd_s

=F3rd_s 159.08 kip =F2nd_s 106.89 kip

≔V3rd_s F3rd_s ≔V2nd_s +F2nd_s F3rd_s

≔Fp_3rd_s_2 ⋅⋅⋅0.2 SDS Ie W3rd_s ≔Fp_3rd_s_3 ⋅⋅⋅0.4 SDS Ie W3rd_s

≔Fp_2nd_s_2 ⋅⋅⋅0.2 SDS Ie W2nd_s ≔Fp_2nd_s_3 ⋅⋅⋅0.4 SDS Ie W2nd_s

≔Fp_3rd_s min ⎛⎝ ,max ⎛⎝ ,Fp_3rd_s_2 Fp_3rd_s_1⎞⎠ Fp_3rd_s_3⎞⎠ =Fp_3rd_s 159.08 kip

≔Fp_2nd_s min ⎛⎝ ,max ⎛⎝ ,Fp_2nd_s_2 Fp_3rd_s_1⎞⎠ Fp_2nd_s_3⎞⎠ =Fp_2nd_s 159.08 kip

Floor shear forces based on N-S direction

≔Fp_3rd_s_NS ――― Fp_3rd_s

346.5 ft =Fp_3rd_s_NS 0.459 ―― kip ft

≔Fp_2nd_s_NS ――― Fp_2nd_s

393 ft =Fp_2nd_s_NS 0.405 ―― kip ft

Lateral transfer capacity

Non-Commercial Use Only

Lateral transfer capacity

Case 1: center structure

Compressive strength ≔f'c 2500 psi

Reinforcement information

Concrete diaphragm shear capacity

≔fy 40 ksi

Slab thickness ≔ts 4.5 in

≔dr 0.5 in ≔Adr ⋅π ―― ⎛⎝dr⎞⎠

Resistance factor for shear ≔ϕv 0.75

Spacing of top steel

Concrete shear capacity

≔sT 8 in

≔ϕVc_dph_c ⋅⋅⋅⋅ϕv 2 ‾‾‾‾‾‾‾⋅f'c 1 psi ts 72 in =ϕVc_dph_c 24.3 kip

≔ϕVc_slab_c ⋅⋅⋅⋅ϕv 2 ‾‾‾‾‾‾‾⋅f'c 1 psi ts DNS_c =ϕVc_slab_c 149.175 kip

Reinforcement shear capacity

≔ϕVs_c ⋅⋅⋅ϕv ―― Adr sT fy 72 in =ϕVs_c 53.014 kip

Non-Commercial Use Only

Maximum limit for shear capacity

≔ϕVn_max_dph_c ⋅⋅⋅⋅ϕv 10 ‾‾‾‾‾‾‾⋅f'c 1 psi ts 72 in =ϕVn_max_dph_c 121.5 kip

≔ϕVn_max_slab_c ⋅⋅⋅⋅ϕv 10 ‾‾‾‾‾‾‾⋅f'c 1 psi ts DNS_c =ϕVn_max_slab_c 745.875 kip

Total shear capacity

≔ϕVn_slab_c min ⎛⎝ ,ϕVn_max_slab_c ϕVc_slab_c⎞⎠

=ϕVn_slab_c 149.175 kip

≔ϕVn_dph_c min ⎛⎝ ,ϕVn_max_dph_c +ϕVs_c ϕVc_dph_c⎞⎠

=ϕVn_dph_c 77.314 kip

Case 2: secondary structure

Concrete shear capacity

≔ϕVc_dph_s ⋅⋅⋅⋅ϕv 2 ‾‾‾‾‾‾‾⋅f'c 1 psi ts 72 in =ϕVc_dph_s 24.3 kip

≔ϕVc_slab_s ⋅⋅⋅⋅ϕv 2 ‾‾‾‾‾‾‾⋅f'c 1 psi ts DNS_s =ϕVc_slab_s 260.55 kip

Reinforcement shear capacity

Non-Commercial Use Only

Reinforcement shear capacity

Reinforcement information

Similar reinforcement to center structure

Maximum limit for shear capacity

≔ϕVn_max_dph_s ⋅⋅⋅⋅ϕv 10 ‾‾‾‾‾‾‾⋅f'c 1 psi ts 72 in =ϕVn_max_dph_s 121.5 kip

≔ϕVn_max_slab_s ⋅⋅⋅⋅ϕv 10 ‾‾‾‾‾‾‾⋅f'c 1 psi ts DNS_s =ϕVn_max_slab_s ⎛⎝ ⋅1.303 103 ⎞⎠ kip

Total shear capacity

≔ϕVn_slab_s min ⎛⎝ ,ϕVn_max_slab_s ϕVc_slab_s⎞⎠

=ϕVn_slab_s 260.55 kip

≔ϕVn_dph_s min ⎛⎝ ,ϕVn_max_dph_s +ϕVs_c ϕVc_dph_s⎞⎠

=ϕVn_dph_s 77.314 kip

Case 3: complete structure

Non-Commercial Use Only

Concrete shear capacity

≔ϕVc_dph ⋅⋅⋅⋅ϕv 2 ‾‾‾‾‾‾‾⋅f'c 1 psi ts 72 in =ϕVc_dph 24.3 kip

≔ϕVc_slab ⋅⋅⋅⋅ϕv 2 ‾‾‾‾‾‾‾⋅f'c 1 psi ts DNS =ϕVc_slab 409.725 kip

Reinforcement information

Similar reinforcement to center structure

Maximum limit for shear capacity

≔ϕVn_max_dph ⋅⋅⋅⋅ϕv 10 ‾‾‾‾‾‾‾⋅f'c 1 psi ts 72 in =ϕVn_max_dph 121.5 kip

≔ϕVn_max_slab ⋅⋅⋅⋅ϕv 10 ‾‾‾‾‾‾‾⋅f'c 1 psi ts DNS =ϕVn_max_slab ⎛⎝ ⋅2.049 103 ⎞⎠ kip

Total shear capacity

≔ϕVn_slab min ⎛⎝ ,ϕVn_max_slab ϕVc_slab⎞⎠

=ϕVn_slab 409.725 kip

≔ϕVn_dph min ⎛⎝ ,ϕVn_max_dph +ϕVs_c ϕVc_dph⎞⎠

=ϕVn_dph 77.314 kip

Diaphragm check based on maximum shear

Non-Commercial Use Only

Diaphragm check based on maximum shear

Case 1: Complete structure

≔Fmax =max ⎛⎝ ,,Fp_roof_NS Fp_3rd_NS Fp_2nd_NS⎞⎠ 0.486 ―― kip ft

Critical complete structure is the 2nd floor Secondary structure

Center structure:

Critical shear is in the center structure ≔VE_2nd 5.3 kip

Critical Moment is in the secondary structure ≔ME_2nd ⋅17.28 kip ft

Slab check

Non-Commercial Use Only

Slab check

≔check_2 =if ⎛⎝ ,,>Fp_2nd ϕVn_slab “not good” “good”⎞⎠ “good”

Slab capacity

≔Slabcap =――― Fp_2nd ϕVn_slab

0.637

Main diaphragm check

≔check_3 =if ⎛⎝ ,,>VE_2nd ϕVn_dph “not good” “good”⎞⎠ “good”

Main diaphragm capacity

≔Dphc_cap =――― VE_2nd ϕVn_dph

0.069

Chord check

≔TU =―――

ME_2nd

72 in

2.88 kip

Collector reinforcement:

≔As =―――

TU

⋅0.9 fy

0.08 in 2

From the beam schedule, beam B204 is acting as the collector

Top reinforcement rebars: number of rebars ≔n 2

Area of rebar ≔As_col ⋅π ――― in

4check minimum beam reinforcement:

≔check_4 =if ⎛⎝ ,,>⋅n As_col As “good” “not good”⎞⎠ “good”

No additional collector reinforcement is required

Diaphragm check based on maximum shear

Non-Commercial Use Only

Case 2: Secondary structure

≔Fmax_s =max ⎛⎝ ,Fp_3rd_s_NS Fp_2nd_s_NS⎞⎠ 0.459 ―― kip ft

Critical secondary structure is the 2nd floor

≔VE_2nd_s 6.4 kip ≔ME_2nd_s ⋅26 kip ft

Slab check

≔check_5 =if ⎛⎝ ,,>Fp_2nd_s ϕVn_slab_s “not good” “good”⎞⎠ “good”

Slab capacity

≔Slabcap_s =――― Fp_2nd_s ϕVn_slab_s

0.611

Main diaphragm check

≔check_6 =if ⎛⎝ ,,>VE_2nd_s ϕVn_dph_s “not good” “good”⎞⎠ “good”

Main diaphragm capacity

≔Dphc_cap_s =――― VE_2nd_s ϕVn_dph_s

0.083

Non-Commercial Use Only

Chord check

≔TU_s =――― ME_2nd_s

72 in

4.333 kip

Collector reinforcement:

≔As_s =――― TU_s

⋅0.9 fy

0.12 in 2

From the beam schedule, beam B204 is acting as the collector

Top reinforcement rebars: number of rebars ≔ns 2

Area of rebar ≔As_col_s ⋅π ――― in

4check minimum beam reinforcement:

≔check_7 =if ⎛⎝ ,,>⋅ns As_col_s As_s “good” “not good”⎞⎠ “good”

No additional collector reinforcement is required

Non-Commercial Use Only

Case 3: Center structure

≔Fmax_c =max ⎛⎝ ,,Fp_roof_c_NS Fp_3rd_c_NS Fp_2nd_s_NS⎞⎠ 0.405 ―― kip ft

Critical center structure is the roof

≔VE_roof_c 3.9 kip ≔ME_roof_c ⋅11.8 kip ft Slab check

≔check_8 =if ⎛⎝ ,,>Fp_roof_c ϕVn_slab_c “not good” “good”⎞⎠ “good”

Slab capacity

≔Slabcap_c =――― Fp_roof_c ϕVn_slab_c

0.673

Main diaphragm check

≔check_9 =if ⎛⎝ ,,>VE_roof_c ϕVn_dph_c “not good” “good”⎞⎠ “good”

Main diaphragm capacity

≔Dphc_cap_c =――― VE_roof_c ϕVn_dph_c

0.05

Non-Commercial Use Only

≔TU_c =――― ME_roof_c

72 in

1.967 kip

Collector reinforcement:

≔As_c =――― TU_c

⋅0.9 fy

0.055 in 2

From the beam schedule, beam B405 is acting as the collector

Top reinforcement rebars:

number of rebars ≔nc 2

Area of rebar ≔As_col_c ⋅π ――― in check minimum beam reinforcement:

≔check_10 =if ⎛⎝ ,,>⋅nc As_col_c As_c “good” “not good”⎞⎠ “good”

No additional collector reinforcement is required

Non-Commercial Use Only

Floor weights, complete structure:

≔Wroof (( ++456.5 336.3 ⋅96 0.5)) kip ≔hroof 30 ft

≔W3rd (( +++1235.5 1046.6 ⋅273 0.5 96.5)) kip ≔h3rd 20 ft

≔W2nd (( +++++1476 1327 9 5 ⋅333 0.5 ⋅273 0.5)) kip ≔h2nd 10 ft

≔W1st (( +++1351 1522 ⋅219 0.5 ⋅333 0.5)) kip ≔h1st 0 ft

Floor weights, center structure Floor weights, secondary structure

≔Wroof_c

++456.5 336.3 ― kip ≔Wroof_s -Wroof Wroof_c

≔W3rd_c

++462 452 ――― +120 96 kip ≔W3rd_s -W3rd W3rd_c

≔W2nd_c

++513 462 ―――― +157 120 kip ≔W2nd_s -W2nd W2nd_c

≔W1st_c

++541 796 ―――― +157 157 kip ≔W1st_s -W1st W1st_c

The base shear, V is given by:

Where; structural height ≔hn hroof

N-S center structure parallel dimension ≔DNS_c 442 in

N-S Secondary structure parallel dimension ≔DNS_s 772 in

N-S Complete structure ≔DNS +DNS_c DNS_s

Non-Commercial Use Only

N-S Fundamental period of vibration

≔Tc ――――― ⋅0.05 hn 1 sec

‾‾‾‾‾‾‾‾‾⋅DNS_c 1 in ≔Ts ―――――

⋅0.05 hn 1 sec

‾‾‾‾‾‾‾‾‾⋅DNS_s 1 in ≔T ―――――

⋅0.05 hn 1 sec

‾‾‾‾‾‾‾‾⋅DNS 1 in

=Tc 0.856 s =Ts 0.648 s =T 0.517 s

≔Cc ――― 0.05

‾‾‾‾3 Tc sec

≔Cs ――― 0.05

‾‾‾‾3 Ts sec

≔C ――― 0.05

‾‾‾‾3 T sec

Horizontal force factor, K ≔K 1

Non-Commercial Use Only

Numerical coefficient of seismic zone ≔Z 2

Non-Commercial Use Only

Total weight of floors

≔W ++Wroof W3rd W2nd

≔Wc ++Wroof_c W3rd_c W2nd_c

≔Ws ++Wroof_s W3rd_s W2nd_s

Base shear of center structure ≔Vc =⋅⋅⋅Z K Cc Wc 313.441 kip

Base shear of secondary structure ≔Vs =⋅⋅⋅Z K Cs Ws 404.449 kip

Base shear of complete structure ≔V =⋅⋅⋅Z K C W 807.072 kip

Force distribution on floors:

≔Ft_c min

,⋅⋅0.004 Vc hn

DNS_c

⋅0.15 Vc

≔Ft_s min

,⋅⋅0.004 Vs hn

DNS_s

⋅0.15 Vs

≔Ft min

,⋅⋅0.004 V hn

DNS

⋅0.15 V

Non-Commercial Use Only

≔sum_c ++⋅Wroof_c hroof ⋅W3rd_c h3rd ⋅W2nd_c h2nd

≔sum_s ++⋅Wroof_s hroof ⋅W3rd_s h3rd ⋅W2nd_s h2nd

≔sum ++⋅Wroof hroof ⋅W3rd h3rd ⋅W2nd h2nd

Force distribution for center structure

≔Froof_c =―――――――― ⋅⋅⎛⎝ -Vc Ft_c⎞⎠ Wroof_c hroof sum_c

138.828 kip ≔Froof_c_dis =―――

Froof_c

264.5 ft

0.525 ―― kip ft

≔F3rd_c =―――――――― ⋅⋅⎛⎝ -Vc Ft_c⎞⎠ W3rd_c h3rd sum_c

112.497 kip ≔F3rd_c_dis =―――

F3rd_c

264.5 ft

0.425 ―― kip ft

≔F2nd_c =―――――――― ⋅⋅⎛⎝ -Vc Ft_c⎞⎠ W2nd_c h2nd sum_c

61.285 kip ≔F2nd_c_dis =―――

F2nd_c

264.5 ft

0.232 ―― kip ft

Force distribution for secondary structure

≔Froof_s =―――――――― ⋅⋅⎛⎝ -Vs Ft_s⎞⎠ Wroof_s hroof sum_s 0 kip

≔F3rd_s =―――――――― ⋅⋅⎛⎝ -Vs Ft_s⎞⎠ W3rd_s h3rd sum_s

241.696 kip ≔F3rd_s_dis =―――

F3rd_s

346.5 ft

0.698 ―― kip ft

≔F2nd_s =―――――――― ⋅⋅⎛⎝ -Vs Ft_s⎞⎠ W2nd_s h2nd sum_s

162.401 kip ≔F2nd_s_dis =―――

F2nd_s

393 ft

0.413 ―― kip ft

Force distribution for complete structure

≔Froof =――――――― ⋅⋅⎛⎝ -V Ft⎞⎠ Wroof hroof sum

190.679 kip ≔Froof_dis =―――

Froof

264.5 ft

0.721 ―― kip ft

≔F3rd =――――――― ⋅⋅⎛⎝ -V Ft⎞⎠ W3rd h3rd sum

380.255 kip ≔F3rd_dis =―――

F3rd

537.5 ft

0.707 ―― kip ft

≔F2nd =――――――― ⋅⋅⎛⎝ -V Ft⎞⎠ W2nd h2nd sum

235.854 kip ≔F2nd_dis =―――

F2nd

537.5 ft

0.439 ―― kip ft

Non-Commercial Use Only

Horizontal force factor ≔Cp 0.1

Wind loads for Albuquerque, NM based on structural height

≔WW 30 psf

Complete structure wind loads ≔WW_roof =⋅⋅WW 264.5 ft 5 ft 39.675 kip

≔WW_3rd =⋅⋅WW 537.5 ft 10 ft 161.25 kip

≔WW_2nd =⋅⋅WW 537.5 ft 10 ft 161.25 kip

Check if wind governs:

Non-Commercial Use Only

Check if wind governs:

≔Check_roof if ⎛⎝ ,,>WW_roof Froof “wind governs” “seismic governs”⎞⎠

=Check_roof “seismic governs”

≔Check_3rd if ⎛⎝ ,,>WW_3rd F3rd “wind governs” “seismic governs”⎞⎠

=Check_3rd “seismic governs”

≔Check_2nd if ⎛⎝ ,,>WW_2nd F2nd “wind governs” “seismic governs”⎞⎠

=Check_2nd “seismic governs” Center structure wind loads

≔WW_roof_c =⋅⋅WW 264.5 ft 5 ft 39.675 kip

≔WW_3rd_c =⋅⋅WW 264.5 ft 10 ft 79.35 kip

≔WW_2nd_c =⋅⋅WW 264.5 ft 10 ft 79.35 kip

Check if wind governs:

≔Check_roof_c if ⎛⎝ ,,>WW_roof_c Froof_c “wind governs” “seismic governs”⎞⎠

=Check_roof_c “seismic governs”

≔Check_3rd_c if ⎛⎝ ,,>WW_3rd_c F3rd_c “wind governs” “seismic governs”⎞⎠

=Check_3rd_c “seismic governs”

≔Check_2nd_c if ⎛⎝ ,,>WW_2nd_c F2nd_c “wind governs” “seismic governs”⎞⎠

=Check_2nd_c “wind governs” Secondary structure wind loads

≔WW_3rd_s =⋅⋅WW 346.5 ft 5 ft 51.975 kip Note: Roof is the critical floor in center structure

≔WW_2nd_s =⋅⋅WW 393 ft 10 ft 117.9 kip

≔Check_3rd_s if ⎛⎝ ,,>WW_3rd_s F3rd_s “wind governs” “seismic governs”⎞⎠ =Check_3rd_s “seismic governs”

≔Check_2nd_s if ⎛⎝ ,,>WW_2nd_s F2nd_s “wind governs” “seismic governs”⎞⎠ =Check_2nd_s “seismic governs”

Lateral capacity according to UBC 64-67

Non-Commercial Use Only

Lateral capacity according to UBC 64-67

Compressive…