https://doi.org/10.14311/APP.2022.33.0300 Acta Polytechnica CTU Proceedings 33:300–308, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague SUGGESTION OF ULTIMATE STRENGTH FORMULAS OF PARTIAL FRAME PILE CAP COMPOSED OF EXTERIOR COLUMN, FOUNDATION BEAM AND PILE Shinji Kishidaa,∗, Tomohisa Mukaib a Shibaura Institute of Technology, 3-7-5 Toyosu Koto-ku Tokyo 135-8548, Japan b Building Research Institute, 1 Tachihara, Tsukuba-shi, Ibaraki-ken 305-0802, Japan ∗ corresponding author: skishida@sic.shibaura-it.ac.jp Abstract. Currently, there is no research and few valid experiments of pile caps. And shear failure mechanism of a pile, exterior column and foundation beam and pile caps in RC structure is not resolved yet under bi-lateral loading. Therefore, a performance evaluation method based on mechanical behaviour for pile caps has not been established. First, the fracture type was specified from the experimental results. Secondary, the ultimate strength formula of the pile cap was proposed based on the previous experimental results. It is a theoretical formula based on the truss-arch theory. It was confirmed that this formula can accurately evaluate the ultimate strength of the pile cap. Keywords: Failure mode, pile-cap, ultimate strength formulas. 1. Introduction Although the seismic performance of buildings after a large earthquake is ensured under the current seismic standards, measures to ensure continuous use after a large earthquake have not been established. Preven- tion of building collapse at the time of a major earth- quake is secured by current earthquake resistance standards but plans to ensure continued use after the earthquake have not been established. It is necessary to develop a method for conducting performance- oriented seismic design with "Sustainability of build- ings after earthquake" as the required [1]. From the viewpoint of continuous usability, it is conceivable that the pile cap will be damaged and deformed in the axial direction will occur and the building will not be able to continue to use due to the inclination of the building. Pile cap is an important structural joint member. Its function is to transfer the stresses occurring on the columns through a group of piles to the ground, taking place the complex stresses under earthquake loading. It is very important to clarify pile cap shear failure mechanism of reinforced con- crete (RC) structures. However, shear failure mecha- nism of a pile, exterior column-beam pile cap in RC structure is not resolved yet under bi-lateral load- ing. Therefore, in this study, the frame experiment of the pile cap was carried out using two types of hoops arranged in the pile cap as experimental fac- tors. The purpose of this study was to clarify the effect of columns and pile cap hoops on pile caps. We evaluated the pile cap shear strength formula pro- posed in the previous study [2, 3] and proposed a pile cap shear strength formula based on the truss-arch mechanism. 2. Outline of Test 2.1. Specimens Fourteen half-scale reinforced concrete pile caps as- sembled a precast pile, an exterior column and aă- foundation beam, those specimens modelled actual middle-high buildings, were tested. Aăconfiguration of specimens, section dimensions and reinforcement details are shown in Figure 1. Specific properties of specimens are summarized in Table 1. Material char- acteristics of concrete and steel are listed in Tableă2, respectively. The constant axial load in compression was applied at the top of the column for all specimens. The depth and width of the column section were 300mm and 300mm, respectively. The depth and width of the foundation beam section were 200mm and 600mm, re- spectively. The length from the center of the column to the loading point on a beam end was 1500mm. The height from the center of the beam to the supporting point on the top of the column or to the bottom sup- port was 1200mm and 1275mm, respectively. Steel pile (Diameter is 190.7mm, thick is 45mm) was used as a precast pile, the embedment length was 100mm, 8-D19 bars were arranged as anchor dowel bars. The grout was filled into the hollow part of the Steel pile for all specimens. All specimens were designed to form shear failure mechanism. For the specimen A-7a and A-8, the pile cap hoop ratio (pcpw) was arranged in 0.22%, the column hoop ratio in pile cap (cpw) was arranged 0.47 times more than the specimen A-7a. For the specimen A-7b and A-9, the pile cap hoop ratio (pcpw) was arranged in 0.10%, the column hoop ratio in pile cap (cpw) was arranged 0.23 times more than the specimen A-7a. For the specimen A-8 and A-9, the column hoop ratio 300 https://doi.org/10.14311/APP.2022.33.0300 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 33/2022 Frame Pile Ultimate Strength Formulas �� � �� � �� � �� �� ���� �� � �� � ��� ��� �� �� �� �� �� �� �� � �� � 3RVLWLYH�ORGLQJ1HJDWLYH�ORGLQJ � 㻌�3LOH�FDS ���� ���� �� �� � �� � �� �� � ��� &RPPRQ�VHFWLRQ ����� �� ��� �� �� �� �� � �� �� �� � ��� 㻼㼕㼘㼑 㻭㼚㼏㼔㼛㼞㼟 㻤㻙㻰㻝㻥㻔㻿㻰㻠㻥㻜㻕 㻯㼛㼘㼡㼙㼚㻌㼙㼍㼕㼚 㼞㼑㼕㼚㼒㼛㼞㼏㼑㼙㼑㼚㼠 㻤㻙㻰㻝㻟 㻌㻌㻌㻔㻿㻰㻟㻠㻡㻕 㻯㼛㼘㼡㼙㼚 㻲㼛㼡㼚㼐㼍㼠㼕㼛㼚㻌㼎㼑㼍㼙 &ROXPQ�ODWHUDO�UHLQIRUFHPHQW '��6'����#�� �,Q�EHDP�FROXPQ�MRLQWV 䚷6'���$#���� 0DLQ�UHLQIRUFHPHQW RI�IRXQGDWLRQ�EHDP ��'���3%6'���� /DWHUDO�UHLQIRUFHPHQW RI�IRXQGDWLRQ�EHDP 8����8UXERQ�#�� 㻌㻌�%DVNHW�W\SH �6SHFLPHQ�$��� $[LDO�IRUFH��&ROXPQ�D[LDO�IRUFH�UDWLR�������� Figure 1. Derails of Specimens. Figure 2. Details of Specimens. 301 Shinji Kishida, Tomohisa Mukai Acta Polytechnica CTU Proceedings Specimen A-7a A-7b A-8 A-9 Axial force (Axial force ratio : 0.2) 500 kN 514 kN 462 kN 467 kN Column Width × Depth 300 mm × 300 mm Main reinforcement 8-D13(SD785) Hoop D6(SD785)@50 Hoop in pile cap (Hoop amount : cPw[%]) D6(SD295A)@100 (0.15) D6(SD295A)@50 (0.30) D6(SD295A)@300 (0.07) D6(SD295A)@300 (0.07) Foundation Beam Width × Depth 200 mm × 600 mm Main reinforcement Upper and Bottom 3-D22(PBSD930) Stirrup U9.0(1275MPa)@50: High-strength shear reinforcement Spacing bar 2-D6(SD295A) Pile Steel pile S45C ϕ190.7 t-45mm Anchor bar 8-D19(SD490) Pile cap Width × Depth × Height 500 mm × 500 mm × 770 mm Vertical reinforcement 4-D6(SD295A) 4-D10(SD295A) Hoop (Hoop amount : pcPw[%]) D6(SD295A)@50 (0.22) D6(SD295A)@100 (0.10) D6(SD295A)@50 (0.22) D6(SD295A)@100 (0.10) 10 0 20 0 20 0 25 4, 5 50 50 ���� 12 0 80 12 0 80 12 0 80 15 4, 5 50 50 ���� 11 7 80 12 0 80 12 0 12 0 13 3 ��� 12 0 10 0 10 0 10 0 10 0 10 0 15 0 ��� Table 1. Properties of Specimens Details. in pile cap (cpw) was arranged in 0.07%. cpw = caw/ (b × l) (1) pcpw = pcaw/ (b × l) (2) Where, b: pile cap width, l: distance between the centers of gravity of the main beams of the foundation beam, cpw, pcpw: the total of each cross-sectional area of the column and the pile cap hoop arranged in the cross section (b × l). 2.2. Loading Apparatus and Instrumentation A loading apparatus is shown in Figure 2. The foun- dation beam end was supported by horizontal roller, while the bottom of pile was supported by a univer- sal joint. The reversed cyclic horizontal load and the constant axial load in compression (an axial load ra- tio of 0.20 in all specimens) were applied at the top of the column through a tri-directional joint by three oil jacks. The jack orthogonal to a horizontal load- ing direction prevented an out-of-plane overturn for specimen. All specimens were controlled by a story drift angle for one loading cycle of 0.25%, two cycle of 0.5%, 1%, 2%, one cycle of 3% respectively, and two cycle of 4%. The story drift angle was defined as a story drift di- vided by height of the column and pile; 3400mm and 2475mm. Lateral force, column axial load and foun- dation beam shear forces were measured by load-cells. Story drift, foundation beam and column deflections, and local displacement of a pile cap panel were mea- sured by displacement transducers. Strains of foun- dation beam bars, column bars and pile cap bars, anchors and hoops were measured by strain gauges. 3. Test results 3.1. Story Shear - Drift Relationship Relationships between the story shear force and the story drift angle are shown in Figure 4. The story shear force was obtained from moment equilibrium between measured beam shear forces and the horizon- tal force at a loading point on the top of the column. The pile cap hoop yielded before the story shear force achieved the maximum strength. For specimen A- 7a and A-7b, the maximum strength was 1.06 times larger on the positive loading than specimen A-7a, 1.11 times larger on negative loading, respectively. After the maximum strength, the decreasing rate was 17% at positive loading for specimen A-7a, 25% at negative loading for specimen A-7b. When the pile cap hoop is larger than the column hoop, the effect 302 vol. 33/2022 Frame Pile Ultimate Strength Formulas Specimen Reinforcing bar Parts used Yield stress! N/mm2" Yield strain [µ] A-7a A-7b D6(SD295A) Column, Pile cap 451.2 2246 D6(SD785)∗ Column 900.4 6684 D10(SD295A) Pile cap 361.0 1989 D13(SD785) Column 816.1 5331 D19(SD490) Anchor 530.0 3027 D22(PBSD930/1080)∗ Main reinforcement of Foundation beam 999.4 6933 U9.0(SBPD1275/1420)∗ Reinforcing bar of Foundation beam 1319.5 8672 A-8 A-9 D6(SD295A)∗ Column, Pile cap 378.7 4079 D6(SD785)∗ Column 928.3 6985 D10(SD295A) Pile cap 362.7 1936 D13(SD785) Column, Pile cap 900.4 6735 D16(SD785)∗ Column 879.1 6716 D19(SD490)∗ Anchor 543.5 3538 D22(PBSD930/1080)∗ Main reinforcement of Foundation beam 1001.6 6990 U9.0(SBPD1275/1420)∗ Reinforcing bar of Foundation beam 1450.8 8507 Table 2. Material Properties of Steel. Specimen Compressive strength! N/mm2" Modulus of elasticity! ×104 N/mm2" Strain at compressive strength [µ] Split tensile strength! N/mm2" A-7a, A-7b 28.4 2.08 2652 2.16 A-8 25.7 2.01 2767 2.21 A-9 26.0 2.02 2755 2.21 Table 3. Material Properties of Concrete. on the maximum shear strength and the ductility ca- pacity were increased. For specimen A-8 and A-9, the smaller the total hoop mass(pcpw +c pw) in the pile cap, the lower the maximum shear strength. 3.2. Crack patterns Crack patterns at the maximum strength are shown in Figure 3. For specimen A-7a and A-8, the pile cap vertical bar and the hoop yielded before the story shear force achieved the maximum strength and the column base was crushed at maximum strength. Af- ter the maximum strength, the pile cap shear crack width did not increase so much, and the damage to the column base became larger. The maximum strength was determined by pile cap shear failure, and after the maximum strength, it was judged that the column was destroyed by crushing. For specimen A- 7b and A-9, before the maximum shear strength, the pile cap hoop, the vertical bar and the column hoop yielded. Due to the width of the pile cap shear cracks has increased after the maximum shear strength, it was judged that pile cap shear failure was destroyed in these two specimens. 4. Consideration of the pile cap hoops 4.1. Pile cap crack properties Specimens A-7a and A-7b in which the total hoop amount (pcpw +c pw) in the pile cap is almost the same and the ratio of pcpw and cpw are different are compared. Comparing the pile cap cracks at the time of the final failure of the two specimens shown in Figure 3, specimen A-7a, which had many pile cap hoops, had dispersed pile cap shear cracks. On the other hand, the specimen A-7b, which had many col- umn hoops, showed a different characteristic that the cracks did not disperse, and the width of several shear cracks increased. This suggests that, of the two types of hoops, the pile cap hoops arranged on the outside contribute a greater shear force and are more effective in preventing brittle shear failure. 4.2. Strain distribution of pile cap hoop 4.2.1. Pile cap hoop Figure 5 shows the strain distribution of pile cap hoops for specimens A-8 and Aŋ9. The two speci- mens have different pcpw, and cpw and have the same amount of reinforcement. At the time of positive 303 Shinji Kishida, Tomohisa Mukai Acta Polytechnica CTU Proceedings ௪௣௖݌ ൅ ௪௖݌ � $��D䚷䚷䚷䚷�5 ��� $��E䚷䚷䚷䚷�5 ��� $��䚷䚷䚷䚷�5 ��� $��䚷䚷䚷䚷�5 ��� 㻯㼛㼙㼜㼞㼑㼟㼟㼕㼛㼚㻌㼒㼍㼕㼘㼡㼞㼑 㼍㼠㻌㼏㼛㼘㼡㼙㼚㻌㼎㼍㼟㼑 㻼㼕㼘㼑㻌㼏㼍㼜㻌㼟㼔㼑㼍㼞㻌㼒㼍㼕㼘㼡㼞㼑 㻯㼛㼙㼜㼞㼑㼟㼟㼕㼛㼚㻌㼒㼍㼕㼘㼡㼞㼑 㼍㼠㻌㼏㼛㼘㼡㼙㼚㻌㼎㼍㼟㼑 㻼㼕㼘㼑㻌㼏㼍㼜㻌㼟㼔㼑㼍㼞㻌㼒㼍㼕㼘㼡㼞㼑 Figure 3. Failure mode at end of test. � ͲϭϮϬ ͲϴϬ ͲϰϬ Ϭ ϰϬ ϴϬ ϭϮϬ Ͳϰ Ͳϯ ͲϮ Ͳϭ Ϭ ϭ Ϯ ϯ ϰ 㻿㼠㼛㼞㼥㻌㼐㼞㼕㼒㼠㻌㼍㼚㼓㼘㼑㻔㻑㻕 㻭㻙㻥 ͲϭϮϬ ͲϴϬ ͲϰϬ Ϭ ϰϬ ϴϬ ϭϮϬ Ͳϰ Ͳϯ ͲϮ Ͳϭ Ϭ ϭ Ϯ ϯ ϰ 㻿 㼠㼛 㼞㼥 㻌㼟 㼔 㼑 㼍 㼞㻌 㼒㼛 㼞㼏 㼑㻌 㻔㼗 㻺 㻕 㻿㼠㼛㼞㼥㻌㼐㼞㼕㼒㼠㻌㼍㼚㼓㼘㼑㻔㻑㻕 㻭㻙㻤 ͲϭϮϬ ͲϴϬ ͲϰϬ Ϭ ϰϬ ϴϬ ϭϮϬ Ͳϰ Ͳϯ ͲϮ Ͳϭ Ϭ ϭ Ϯ ϯ ϰ 㻿㼠㼛㼞㼥㻌㼐㼞㼕㼒㼠㻌㼍㼚㼓㼘㼑㻔㻑㻕 㻭㻙㻣㼎 ͲϭϮϬ ͲϴϬ ͲϰϬ Ϭ ϰϬ ϴϬ ϭϮϬ Ͳϰ Ͳϯ ͲϮ Ͳϭ Ϭ ϭ Ϯ ϯ ϰ 㻿 㼠㼛 㼞㼥 㻌㼟 㼔 㼑 㼍 㼞㻌 㼒㼛 㼞㼏 㼑 㻌㻔 㼗㻺 㻕 㻿㼠㼛㼞㼥㻌㼐㼞㼕㼒㼠㻌㼍㼚㼓㼘㼑㻔㻑㻕 㻭㻙㻣㼍 DĂdžŝŵƵŵ ƐƚƌĞŶŐƚŚ ^ŚĞĂƌ ĐƌĂĐŬ ,ŽŽƉ�LJŝĞůĚ� ŝŶ�ƉŝůĞ�ĐĂƉ ,ŽŽƉ�LJŝĞůĚ�ŝŶ�ĐŽůƵŵŶ 4PD[ �����N1� 4PLQ �����N1� 4PD[ �����N1� 4PLQ �����N1� 4PD[ ����N1� 4PLQ �����N1�4PD[ ����N1 4PLQ �����N1� î��0D[LPXP�VWUHQJWK��������ᶭ��6KHDU�FUDFN�LQ�SLOH�FDS� �ࠐ�+RRS�\LHOG�LQ�SLOH�FDS��Ƒ��&ROXPQ�+RRS�\LHOG�LQ�SLOH�FDSۍ����$QFKRU�\LHOG� Figure 4. Details of Specimens. � ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯй ϭй WŝůĞ�ĐĂƉ�ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ� ƐŝĚĞ &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯй ϭй WŝůĞ�ĐĂƉ�ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ� ďŽƚƚŽŵ�ƐŝĚĞ &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯйϭй WŝůĞ�ĐĂƉ�ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ�ƐŝĚĞ� &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ zŝ Ğů Ě Ɛƚ ƌĂ ŝŶ Ϭ ϭϬϬ ϮϬϬ ϯϬϬ ϰϬϬ ϱϬϬ ϲϬϬ ϳϬϬ ϴϬϬ ϵϬϬ 'Ă ƵŐ Ğ� ƉŽ Ɛŝƚ ŝŽ Ŷ; ŵ ŵ Ϳ ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯйϭй WŝůĞ�ĐĂƉ�ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ�ƐŝĚĞ� &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ ����������������������������������������� ����������������������������������������� ���������� ����������������������������� ����������������������������������������� $���� �QHJDWLYH�ORDGLQJ�� $��� ��QHJDWLYH�ORDGLQJ��$���� �SRVLWLYH�ORDGLQJ�� $��� �SRVLWLYH�ORDGLQJ�� Figure 5. Strain distribution of pile cap hoop. 304 vol. 33/2022 Frame Pile Ultimate Strength Formulas ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱ й Ϯ й ϭ й zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ WŝůĞ�ĐĂƉ� ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ�ƐŝĚĞ� &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵƉƉĞƌ� ďĞĚ ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯ й ϭ й zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ WŝůĞ�ĐĂƉ� ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ�ƐŝĚĞ� &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵƉƉĞƌ�ďĞĚ ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯй ϭй zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ WŝůĞ�ĐĂƉ� ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ�ƐŝĚĞ� &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵƉƉĞƌ�ďĞĚ Ϭ ϭϬϬ ϮϬϬ ϯϬϬ ϰϬϬ ϱϬϬ ϲϬϬ ϳϬϬ ϴϬϬ ϵϬϬ ϭϬϬϬ ϭϭϬϬ 'Ă ƵŐ Ğ� ƉŽ Ɛŝƚ ŝŽ Ŷ; ŵ ŵ Ϳ ^ƚƌĂŝŶ;ʅͿ Ϭ͘ϱй Ϯй ϭй zŝ Ğů Ě� Ɛƚ ƌĂ ŝŶ WŝůĞ�ĐĂƉ� ƵƉƉĞƌ�ƐŝĚĞ� WŝůĞ�ĐĂƉ�ďŽƚƚŽŵ�ƐŝĚĞ� &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵŶĚĞƌ�ďĞĚ &ŽƵŶĚĂƚŝŽŶ�ďĞĂŵ�ƵƉƉĞƌ�ďĞĚ �������������������������������������������������������������������������� �������������������������������������� �������������������������������������� $��E��SRVLWLYH�ORDGLQJ� $��E� ��QHJDWLYH�ORDGLQJ�� $����SRVLWLYH�ORDGLQJ�� $����QHJDWLYH�ORDGLQJ� Figure 6. Strain distribution of column hoops in pile cap. loading, the strain at the lower end of the foundation beam was large in both specimens. For specimen A- 8, the strain did not reach the yield to R = 1%, but for specimen A-9, the strain had reached the yield to R = 1%. It is considered that the tensile force acting one hoop was dispersed in the specimen A-8 with a large amount of pcpw, and the hoop did not yield until R=2%. On the negative loading, the strain amount of specimen A-8 was smaller than that of the positive loading until the maximum shear strength (R = −1%). Also, the strain at the top of the pile cap was larger than that at the bottom of the foun- dation beam, and the difference depending on the loading direction was observed. In specimen A-9, the strain became smaller as compared to the pos- itive loading. Compared with test specimen A-8, the strain increased up to R = −1% as in the case of positive loading. As described above, by arranging a large amount of pcpw, it is possible to reduce the ratio of a single hoop to the shear force acting on the pile cap. 4.2.2. Column hoop in pile cap Figure 6 shows the strain distribution of the column hoops in the pile cap of specimen A-7b and speci- men A-4. In test specimen A-4, cpw was arranged 0.15%, and pcpw was recombined in the same amount as specimen A-7b. At the time of positive loading, the strain at the lower end of the foundation beam tended to increase as in the pile cap hoops in both specimens, but the column hoops did not yield even at the maximum strength (R = 2%). When the strain values of the two specimens were compared, the strain value of the column hoop was almost unchanged even if cpw was increased. The ratio of the shear force act- ing on the pile cap differs between the pile cap hoop and the column hoop arranged in the pile cap. It is considered that pile cap hoops contribute more effec- tively to shear resistance because the tensile force of pile cap hoops decreases with increasing. 4.3. Relationship between the pile cap input shear force and the amount of hoop Figure 7 shows the relationship between the maxi- mum pile cap input shear force, the amount of pile cap hoop, and the pile cap hoop in the test specimen of this study and specimen A-4. The tensile strength of the main bar of the foundation beam was calcu- lated from the strain at the critical section position of the foundation beam, and the input shear force of the pile cap was calculated by the following equation. Vj = T − Qc (6) Where, T : tensile force of foundation beam bar, Qc: story shear force. When cpw was less than 0.15%, the input shear force increased as cpw increased, but when cpw was more than 0.15%, the input shear force became al- most constant. The effect of cpw on pile cap strength was considered to be limited to cpw ≤ 0.15% in this study. Even when pcpw increased, the input shear force showed almost the same value or a tendency to slightly increase. At the time of positive loading, the input shear force generally tends to increase as the to- tal hoop amount in the pile cap increases. However, the relationship between the total hoop amount and the input shear force became constant under negative loading. 4.4. Carrying of the pile cap hoop Figure 8 shows the ratio of the average stress and the yield stress of the pile cap hoop at the maximum strength in the specimen of this study and specimen A-4. In the case of pile cap hoops, the stress at the maximum shear strength was large in all specimens at positive loading. On the other hand, at the time of negative loading, the stress varies greatly for each specimen. For the column hoops, the stress at pos- itive loading was smaller than that of the pile cap hoops. Furthermore, the carrying stress of the col- umn hoop was almost constant even if cpw was ar- ranged more than 0.15%. This is consistent with the 305 Shinji Kishida, Tomohisa Mukai Acta Polytechnica CTU Proceedings � ��� ��� ��� ��� ��� ��� ��� ��� ��� F3Z�SF3Z��� sũ;ŬEͿ � ��� ��� ��� ��� ��� � ��� ��� ��� SF3Z��� sũ;ŬEͿ 䕕䠖ĐWǁсϬ͘Ϭϳй 䕧䠖ĐWǁсϬ͘ϭϱй � ��� ��� ��� ��� ��� � ��� ��� ��� F3Z��� sũ;ŬEͿ 䕕䠖ƉĐWǁсϬ͘ϭй 䕧䠖ƉĐWǁсϬ͘ϮϮй �D��Vj ��cPw �E��Vj ��pcPw �F��Vj ��cPw�pcPw� Figure 7. Relationship between maximum input shear force and two kinds of hoop in pile cap. Ϭ Ϭ͘Ϯ Ϭ͘ϰ Ϭ͘ϲ Ϭ͘ϴ ϭ ϭ͘Ϯ Ϭ Ϭ͘ϭ Ϭ͘Ϯ Ϭ͘ϯ t Žƌ Ŭŝ ŶŐ �Ɛƚ ƌĞ ƐƐ ͬz ŝĞ ůĚ �Ɛƚ ƌĞ ƐƐ ƉĐWǁ;йͿ WŝůĞ�ĐĂƉ�ŚŽŽƉ Ϭ Ϭ͘Ϯ Ϭ͘ϰ Ϭ͘ϲ Ϭ͘ϴ ϭ ϭ͘Ϯ Ϭ Ϭ͘ϭ Ϭ͘Ϯ Ϭ͘ϯ t Žƌ Ŭŝ ŶŐ �Ɛƚ ƌĞ ƐƐ ͬz ŝĞ ůĚ �Ɛƚ ƌĞ ƐƐ ĐWǁ;йͿ �ŽůƵŵŶ�ŚŽŽƉ SFQF ����� SFQR ����� FQR ����� FQF ����� FQF ���F3Z� &RORXU���� &ORVLQJ�GLUHFWLRQ�� 2SHQLQJ�GLUHFWLRQ� Figure 8. Ratio of the average stress and the yield stress of the pile cap hoop. 1. when ν0σB −c σt < 0 Vu is smaller of the follow. Vu = # $% $& λc ν0 σB + cn cpwe cσwy 3 cbe cje λc ν0 σB 2 cbe cje (3) 2. when ν0σB −c σt ≥ 0 and ν0σB −c σt −pc σt < 0 cVt = 2 cn cpwe cσwy cbe cje pcVt is smaller of the follow. pcVt = # $$$% $$$& λpc ( ν0 σB −c σt) + pcn pcpwe pcσwy 3 pcbe pcje λpc ( ν0 σB −c σt) 2 pcbe pcje (4) Vu =c Vt +pc Vt 3. when ν0σB −c σt −pc σt ≥ 0 cVt = 2 cn cpwe cσwy cbe cje pcVt = 2 pcn pcpwe pcσwy pcbe pcje Va = (ν0σB −c σt −pc σt) bxn 2 sin 2θ (5) Vu =c Vt +pc Vt + Va Note: xn = D 4 (1 + 2η), θ = tan−1 D−xn L , ν0 = 2.3 σ−0.33 B . Symbols are explained in the nomenclature. Table 4. Pile cap shear strength formula based on previous investigation. 306 vol. 33/2022 Frame Pile Ultimate Strength Formulas Loading direction Closing Opening Closing Opening Carrying ratio pcnc pcno cnc cno 0.96 0.64 5.4 cPw (cPw < 0.15 %) 0.81 (cPw > 0.15 %) 0.59 Table 5. Carrying stress coefficient. �D��RSHQLQJ�GLUHFWLRQ� �E��FORVLQJ�GLUHFWLRQ� Ϭ ϮϱϬ ϱϬϬ ϳϱϬ ϭϬϬϬ Ϭ ϮϱϬ ϱϬϬ ϳϱϬ ϭϬϬϬ sƵ͗ ƉŝůĞ�ĐĂƉ�ŝŶƉƵƚ ƐŚĞĂƌ�ĨŽƌĐĞ;ŬEͿ YƵ͗ �ĂůĐƵůĂƚŝŽŶ�ƌĞƐƵůƚ�;ŬEͿ ĂǀĞƌĂŐĞ͗ϭ͘Ϭϭ �ŽĞĨĨŝĐŝĞŶƚ�ŽĨ�ǀĂƌŝĂƚŝŽŶ͗Ϭ͘ϭϲϲ Ϭ ϮϱϬ ϱϬϬ ϳϱϬ ϭϬϬϬ Ϭ ϮϱϬ ϱϬϬ ϳϱϬ ϭϬϬϬ sƵ͗ ƉŝůĞ�ĐĂƉ�ŝŶƉƵƚ ƐŚĞĂƌ�ĨŽƌĐĞ;ŬEͿ YƵ͗ �ĂůĐƵůĂƚŝŽŶ�ƌĞƐƵůƚ�;ŬEͿ ĂǀĞƌĂŐĞ͗ϭ͘Ϭϲ �ŽĞĨĨŝĐŝĞŶƚ�ŽĨ�ǀĂƌŝĂƚŝŽŶ͗Ϭ͘ϭϱϰ Figure 9. Relationship between experimental values and proposed calculations. fact that the input shear force became almost con- stant in the range of cpw over 0.15% in the relation- ship between Vj and cpw. Based on the above results, the values obtained by approximately calculating the ratio of the carrying stress applied to the hoops of the pile cap and the column are shown by the bro- ken line in Figure 8. And Table 5 shows the carrying stress coefficient obtained from the Figure 8. Table 4 shows pile cap shear strength formula applied carry- ing stress coefficient. The stress of the hoops under positive loading was approximated in the range of cpw < 0.15% so that the proportion of the hoops increased in proportion to the increase of cpw. In the range of ăcpw ≥ ă0.15%, based on the fact that the strength did not change even if cpw was increased, using the ratio of the mate- rial strength of specimen A-7a and the carrying stress of 0.81, the upper limit of cpw ·c σy was set to 0.55 (N/mm2). 5. 5. Compatibility of pile cap ultimate shear strength formula The calculation was performed by substituting the co- efficients obtained in Table 5. The target specimens were the specimens of this study and the specimens judged to be pile cap shear failure in past experiments [2–4]. Figure 9 shows the comparison between the calculation result by the proposed formula and the experimental value. The input shear force was deter- mined from the value of the strain gauge by defining the position at which the strain of the main bar of the foundation beam was maximum as the critical section. As shown in Figure 9, the experimental / calculated values obtained by the proposed formula were generally within ±20% (dotted line in the Fig- ure), and the average and coefficient of variation were 1.01-1.06 and 15.4-16.6%, respectively. The average (the dashed line in the Figure) and the standard devi- ation were also considered to be valid as experimen- tal values. However, only the standard type speci- men was greatly underestimated on the side where the column-foundation beam opened. The cause is considered to be that the effective reinforcement ra- tio of the pile cap is extremely low at 0.03%, and the truss mechanism has not been formed. From this experiment, the minimum reinforcement amount is set to 0.07% as the applicable range of the pile cap effective reinforcement ratio. 6. Conclusions 1. Among the reinforcements arranged in the pile cap, the pile cap stirrups contributed more to the shear load. By arranging many pile cap hoops, an in- crease in pile cap shear crack width was suppressed. 2. It was confirmed that the relationship between the total amount of hoops in the pile cap and the input shear force to the pile cap was different depending on the loading direction. 3. The shear strength formula proposed in the past 307 Shinji Kishida, Tomohisa Mukai Acta Polytechnica CTU Proceedings was able to be evaluated safely by considering the load ratio of the hoops at the time of pile cap shear failure. Acknowledgements This work was supported by JSPS KAKENHI Grant Number JP23560679. This experiment was carried out in Structural Lab. at BRI. Authors would like to express their gratitude to JAPAN PILE Co., Ltd. for provid- ing steel piles, Neturen Co., Ltd. and TOKYO TEKKO CO., Ltd. for providing steel bars, we are deeply grateful to them. List of symbols cbe the truss effective width in column cje the truss effective depth in column cVt column shear strength in truss mechanism pcpwe the effective ratio of shear reinforcing bar in the pile cap pcVt pile cap shear strength in truss mechanism cσt column compression stress in truss mechanism pcσt pile cap compression stress in truss mechanism cpwe the effective ratio of shear reinforcing bar in the column cσwy the yield stress in column reinforcing bar cσwy truss effective coefficient in column pcσwy the yield stress in pile cap reinforcing bar pcbe the truss effective width in pile cap pcje the truss effective depth in pile cap b pile cap effective width D pile cap effective depth L member length Va shear strength in arch mechanism xn the neutral axis position in arch mechanism η axial force ratio θ the angle of compression strut in the arch mechanism λc the yield stress in column reinforcing bar λpc the truss effective coefficient in pile cap ν0 effective coefficient of concrete compression stress σa compression stress in arch mechanism σB concrete compression stress References [1] Mukai T., Kikitsu H., Morita K., and Fukuyama H.: Factor Analysis of barriers to Post-Earthquake Functionality for Buildings (Part.1-5), Summaries of Technical Papers of Annual Meeting, Architectural Institute of Japan pp. 445-446, 2014. [2] S. Kishida, T. Mukai. Experimental study on the reinforced concrete pile-cap with a pil, exterior column and foundation beam, fib 2018 Congress in Australia, ID302, 2018. [3] S. Kishida S, T. Mukai, H. Watanabe. Study on structural performance evaluation for concrete pile system with post-earthquake functional use (Part.29), Summaries of Technical Papers of Annual Meeting, Architectural Institute of Japan pp. 221-222, 2019. [4] S. Kishida, T. Mukai, Y. Maida. A study on Failure mode of pile caps on the exterior frame with precast pile, Summaries of Technical Papers of Annual Convention of Japan Concrete Institute 42(2):271-276, 2019. 308